From bb11e7ae32284c682bcf93e65466ea10418b4bda Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Tue, 12 Dec 2023 04:10:29 +0000
Subject: [PATCH 01/21] begun tracer_conservative transport

---
 gusto/active_tracers.py    |   7 ++-
 gusto/common_forms.py      |  25 +++++++-
 gusto/configuration.py     |   2 +
 gusto/equations.py         | 117 ++++++++++++++++++++++++++++++++++++-
 gusto/transport_methods.py |  12 ++++
 5 files changed, 160 insertions(+), 3 deletions(-)

diff --git a/gusto/active_tracers.py b/gusto/active_tracers.py
index ab0d59b28..1959deedb 100644
--- a/gusto/active_tracers.py
+++ b/gusto/active_tracers.py
@@ -53,6 +53,7 @@ class ActiveTracer(object):
     """
     def __init__(self, name, space, variable_type,
                  transport_eqn=TransportEquationType.advective,
+                 density_name=None,
                  phase=Phases.gas, chemical=None):
         """
         Args:
@@ -62,7 +63,10 @@ def __init__(self, name, space, variable_type,
                 the type of tracer variable (e.g. mixing ratio or density).
             transport_eqn (:class:`TransportEquationType`, optional): enumerator
                 indicating the type of transport equation to be solved (e.g.
-                advective). Defaults to `TransportEquationType.advective`.
+                advective). Defaults to `TransportEquationType.advective`.#
+            density_name (str): the name of the associated density for a mixing
+                ratio when using the tracer_conservative transport. 
+                Defaults to None.
             phase (:class:`Phases`, optional): enumerator indicating the phase
                 of the tracer variable. Defaults to `Phases.gas`.
             chemical (str, optional): string to describe the chemical that this
@@ -75,6 +79,7 @@ def __init__(self, name, space, variable_type,
         self.name = name
         self.space = space
         self.transport_eqn = transport_eqn
+        self.density_name = density_name
         self.variable_type = variable_type
         self.phase = phase
         self.chemical = chemical
diff --git a/gusto/common_forms.py b/gusto/common_forms.py
index 67e3b1e96..d1355a44a 100644
--- a/gusto/common_forms.py
+++ b/gusto/common_forms.py
@@ -12,7 +12,7 @@
 __all__ = ["advection_form", "continuity_form", "vector_invariant_form",
            "kinetic_energy_form", "advection_equation_circulation_form",
            "diffusion_form", "linear_advection_form", "linear_continuity_form",
-           "split_continuity_form"]
+           "split_continuity_form", "tracer_conservative_form"]
 
 
 def advection_form(test, q, ubar):
@@ -262,3 +262,26 @@ def split_continuity_form(equation):
                 map_if_true=drop)
 
     return equation
+    
+def tracer_conservative_form(test, q, ubar):
+    u"""
+    The form corresponding to the continuity transport operator.
+
+    This describes ∇.(u*q) for transporting velocity u and a 
+    transported tracer, q.
+
+    Args:
+        test (:class:`TestFunction`): the test function.
+        q (:class:`ufl.Expr`): the tracer to be transported. If this
+        is a mixing ratio, q = m*rho. If it is a density, q = rho.
+        ubar (:class:`ufl.Expr`): the transporting velocity.
+
+    Returns:
+        class:`LabelledForm`: a labelled transport form.
+    """
+    
+    L = inner(test, div(outer(q, ubar)))*dx
+    form = transporting_velocity(L, ubar)
+    
+    return transport(form, TransportEquationType.tracer_conservative)
+    
diff --git a/gusto/configuration.py b/gusto/configuration.py
index 3694ae9ea..b013c2a7c 100644
--- a/gusto/configuration.py
+++ b/gusto/configuration.py
@@ -31,6 +31,7 @@ class TransportEquationType(Enum):
     conservative: ∂q/∂t + ∇.(u*q) = 0                                         \n
     vector_invariant: ∂q/∂t + (∇×q)×u + (1/2)∇(q.u) + (1/2)[(∇q).u -(∇u).q)] = 0
     circulation: ∂q/∂t + (∇×q)×u + non-transport terms = 0
+    tracer_conservative: to fill in.
     """
 
     no_transport = 702
@@ -38,6 +39,7 @@ class TransportEquationType(Enum):
     conservative = 291
     vector_invariant = 9081
     circulation = 512
+    tracer_conservative = 296
 
 
 class Configuration(object):
diff --git a/gusto/equations.py b/gusto/equations.py
index bd61e795b..1cb42228b 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -20,7 +20,8 @@
 from gusto.common_forms import (
     advection_form, continuity_form, vector_invariant_form,
     kinetic_energy_form, advection_equation_circulation_form,
-    diffusion_form, linear_continuity_form, linear_advection_form
+    diffusion_form, linear_continuity_form, linear_advection_form,
+    tracer_conservative_form
 )
 from gusto.active_tracers import ActiveTracer, Phases, TracerVariableType
 from gusto.configuration import TransportEquationType
@@ -429,6 +430,42 @@ def add_tracers_to_prognostics(self, domain, active_tracers):
             else:
                 raise TypeError(f'Tracers must be ActiveTracer objects, not {type(tracer)}')
 
+    def generate_tracer_mass_terms(self):
+        """
+        Builds the weak time derivative terms for the equation set.
+
+        Generates the weak time derivative terms ("mass terms") for all the
+        prognostic variables of the equation set.
+        
+        The mass terms can differ depending on the tracer type. A mixing ratio
+        that is being transported conservatively will need its mass form multiplied
+        
+
+        Returns:
+            :class:`LabelledForm`: a labelled form containing the mass terms.
+        """
+
+        for _, tracer in enumerate(active_tracers):
+            idx = self.field_names.index(tracer.name)
+            tracer_prog = split(self.X)[idx]
+            tracer_test = self.tests[idx]
+            if tracer.transport_eqn == TransportEquationType.tracer_conservative:
+                if tracer.variable_type == TracerVariableType.density:
+                    mass = subject(prognostic(inner(prog, test)*dx, field_name), self.X)
+                elif tracer.variable_type == TracerVariableType.mixing_ratio:
+                    ref_density_idx = self.field_names.index(tracer.density_name)
+                    ref_density = split(self.X)[ref_density_idx]
+                    q = tracer_prog*ref_density
+                    mass = subject(prognostic(inner(q, test)*dx, field_name), self.X)
+                else:
+                    ValueError(f'Tracer type {tracer.tracer_type} is not currently implemented with tracer_conservative transport')
+            if i == 0:
+                mass_form = time_derivative(mass)
+            else:
+                mass_form += time_derivative(mass)
+
+        return mass_form
+
     def generate_tracer_transport_terms(self, active_tracers):
         """
         Adds the transport forms for the active tracers to the equation set.
@@ -472,6 +509,21 @@ def generate_tracer_transport_terms(self, active_tracers):
                     tracer_adv = prognostic(
                         continuity_form(tracer_test, tracer_prog, u),
                         tracer.name)
+                elif tracer.transport_eqn == TransportEquationType.tracer_conservative:
+                    if tracer.variable_type == TracerVariableType.density:
+                        tracer_adv = prognostic(
+                            tracer_conservative_form(tracer_test, tracer_prog, u),
+                            tracer.name)
+                    elif tracer.variable_type == TracerVariableType.mixing_ratio:
+                        ref_density_idx = self.field_names.index(tracer.density_name)
+                        ref_density = split(self.X)[ref_density_idx]
+                        q = tracer_prog*ref_density
+                        tracer_adv = prognostic(
+                            tracer_conservative_form(tracer_test, q, u),
+                            tracer.name)
+                    else:
+                        ValueError(f'Tracer type {tracer.tracer_type} is not currently implemented with tracer_conservative transport')
+                    
                 else:
                     raise ValueError(f'Transport eqn {tracer.transport_eqn} not recognised')
 
@@ -567,6 +619,69 @@ def __init__(self, domain, active_tracers, Vu=None):
 
         # Add transport of tracers
         self.residual += self.generate_tracer_transport_terms(active_tracers)
+        
+class ConservativeCoupledTransportEquation(PrognosticEquationSet):
+    u"""
+    Discretises the transport equation,               \n
+    ∂q/∂t + (u.∇)q = F,
+    with the application of active tracers.
+    As there are multiple tracers or species that are
+    interacting, q and F are vectors.
+    This equation can be enhanced through the addition of
+    sources or sinks (F) by applying it with physics schemes.
+    This takes in tracers that might obey different forms
+    of the transport equation (i.e. advective, conservative)
+    but will evolve all the fields in a conservative manner.
+    """
+    def __init__(self, domain, active_tracers, Vu=None):
+        """
+        Args:
+            domain (:class:`Domain`): the model's domain object, containing the
+                mesh and the compatible function spaces.
+            active_tracers (list): a list of `ActiveTracer` objects
+                that encode the metadata for any active tracers to be included
+                in the equations. This is required for using this class; if there
+                is only a field to be advected, use the AdvectionEquation
+                instead.
+            Vu (:class:`FunctionSpace`, optional): the function space for the
+                velocity field. If this is not specified, uses the HDiv spaces
+                set up by the domain. Defaults to None.
+        """
+
+        self.active_tracers = active_tracers
+        self.terms_to_linearise = {}
+        self.field_names = []
+        self.space_names = {}
+
+        # Build finite element spaces
+        self.spaces = []
+
+        # Add active tracers to the list of prognostics
+        if active_tracers is None:
+            active_tracers = []
+        self.add_tracers_to_prognostics(domain, active_tracers)
+
+        # Make the full mixed function space
+        W = MixedFunctionSpace(self.spaces)
+
+        full_field_name = "_".join(self.field_names)
+        PrognosticEquation.__init__(self, domain, W, full_field_name)
+
+        if Vu is not None:
+            domain.spaces.add_space("HDiv", Vu, overwrite_space=True)
+        V = domain.spaces("HDiv")
+        _ = self.prescribed_fields("u", V)
+
+        self.tests = TestFunctions(W)
+        self.X = Function(W)
+        
+        mass_form = self.generate_tracer_mass_terms()
+
+        self.residual = subject(mass_form, self.X)
+
+        # Add transport of tracers
+        self.residual += self.generate_tracer_transport_terms(active_tracers)        
+
 
 
 # ============================================================================ #
diff --git a/gusto/transport_methods.py b/gusto/transport_methods.py
index 4fc951b6d..c646893d1 100644
--- a/gusto/transport_methods.py
+++ b/gusto/transport_methods.py
@@ -174,6 +174,18 @@ def __init__(self, equation, variable, ibp=IntegrateByParts.ONCE,
                 raise NotImplementedError('Outflow not implemented for upwind vector invariant')
             form = upwind_vector_invariant_form(self.domain, self.test, self.field, ibp=ibp)
 
+        elif self.transport_equation_type == TransportEquationType.tracer_conservative:
+            # Can use the conservative equation, but ensuring the correct field
+            # for a mixing ratio.
+            if self.field.variable_type == TracerVariableType.density:
+                form = upwind_conservative_form(self.domain, self.test, self.field, ibp=ibp)
+            elif self.field.variable_type == TracerVariableType.mixing_ratio:
+                ref_density_idx = self.eqn.field_names.index(tracer.density_name)
+                ref_density = split(self.X)[ref_density_idx]
+                q = self.field*ref_density
+                form = upwind_conservative_form(self.domain, self.test, q, ibp=ibp)
+        
+            form = upwind_conservative_form(self.domain, self.test, self.field, ibp=ibp)
         else:
             raise NotImplementedError('Upwind transport scheme has not been '
                                       + 'implemented for this transport equation type')

From e6bb82741183899b6b576a329e94d94c10a7bc94 Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Tue, 12 Dec 2023 21:58:36 +0000
Subject: [PATCH 02/21] DGUpwind for tracer_conservative

---
 gusto/active_tracers.py    |  3 +++
 gusto/configuration.py     |  3 ++-
 gusto/equations.py         | 33 +++++++++++----------------------
 gusto/transport_methods.py | 29 +++++++++++++++++------------
 4 files changed, 33 insertions(+), 35 deletions(-)

diff --git a/gusto/active_tracers.py b/gusto/active_tracers.py
index 1959deedb..93d0a1e73 100644
--- a/gusto/active_tracers.py
+++ b/gusto/active_tracers.py
@@ -87,6 +87,9 @@ def __init__(self, name, space, variable_type,
         if (variable_type == TracerVariableType.density and transport_eqn == TransportEquationType.advective):
             logger.warning('Active tracer initialised which describes a '
                            + 'density but solving the advective transport eqn')
+                           
+        if (transport_eqn == TransportEquationType.tracer_conservative and density_name == None):
+            raise ValueError(f'Active tracer {name} using tracer conservative transport needs an associated density.')
 
 
 class WaterVapour(ActiveTracer):
diff --git a/gusto/configuration.py b/gusto/configuration.py
index b013c2a7c..b2a3acfa0 100644
--- a/gusto/configuration.py
+++ b/gusto/configuration.py
@@ -31,7 +31,8 @@ class TransportEquationType(Enum):
     conservative: ∂q/∂t + ∇.(u*q) = 0                                         \n
     vector_invariant: ∂q/∂t + (∇×q)×u + (1/2)∇(q.u) + (1/2)[(∇q).u -(∇u).q)] = 0
     circulation: ∂q/∂t + (∇×q)×u + non-transport terms = 0
-    tracer_conservative: to fill in.
+    tracer_conservative: to fill in. for transporting a tracer that obeys an
+    advection equation in a conservative manner.
     """
 
     no_transport = 702
diff --git a/gusto/equations.py b/gusto/equations.py
index 1cb42228b..ab39733cd 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -450,15 +450,11 @@ def generate_tracer_mass_terms(self):
             tracer_prog = split(self.X)[idx]
             tracer_test = self.tests[idx]
             if tracer.transport_eqn == TransportEquationType.tracer_conservative:
-                if tracer.variable_type == TracerVariableType.density:
-                    mass = subject(prognostic(inner(prog, test)*dx, field_name), self.X)
-                elif tracer.variable_type == TracerVariableType.mixing_ratio:
-                    ref_density_idx = self.field_names.index(tracer.density_name)
-                    ref_density = split(self.X)[ref_density_idx]
-                    q = tracer_prog*ref_density
-                    mass = subject(prognostic(inner(q, test)*dx, field_name), self.X)
-                else:
-                    ValueError(f'Tracer type {tracer.tracer_type} is not currently implemented with tracer_conservative transport')
+                ref_density_idx = self.field_names.index(tracer.density_name)
+                ref_density = split(self.X)[ref_density_idx]
+                q = tracer_prog*ref_density
+                mass = subject(prognostic(inner(q, test)*dx, field_name), self.X)
+                
             if i == 0:
                 mass_form = time_derivative(mass)
             else:
@@ -510,19 +506,12 @@ def generate_tracer_transport_terms(self, active_tracers):
                         continuity_form(tracer_test, tracer_prog, u),
                         tracer.name)
                 elif tracer.transport_eqn == TransportEquationType.tracer_conservative:
-                    if tracer.variable_type == TracerVariableType.density:
-                        tracer_adv = prognostic(
-                            tracer_conservative_form(tracer_test, tracer_prog, u),
-                            tracer.name)
-                    elif tracer.variable_type == TracerVariableType.mixing_ratio:
-                        ref_density_idx = self.field_names.index(tracer.density_name)
-                        ref_density = split(self.X)[ref_density_idx]
-                        q = tracer_prog*ref_density
-                        tracer_adv = prognostic(
-                            tracer_conservative_form(tracer_test, q, u),
-                            tracer.name)
-                    else:
-                        ValueError(f'Tracer type {tracer.tracer_type} is not currently implemented with tracer_conservative transport')
+                    ref_density_idx = self.field_names.index(tracer.density_name)
+                    ref_density = split(self.X)[ref_density_idx]
+                    q = tracer_prog*ref_density
+                    tracer_adv = prognostic(
+                        tracer_conservative_form(tracer_test, q, u),
+                        tracer.name)
                     
                 else:
                     raise ValueError(f'Transport eqn {tracer.transport_eqn} not recognised')
diff --git a/gusto/transport_methods.py b/gusto/transport_methods.py
index c646893d1..c890bd261 100644
--- a/gusto/transport_methods.py
+++ b/gusto/transport_methods.py
@@ -4,7 +4,7 @@
 
 from firedrake import (
     dx, dS, dS_v, dS_h, ds_t, ds_b, ds_v, dot, inner, outer, jump,
-    grad, div, FacetNormal, Function, sign, avg, cross, curl
+    grad, div, FacetNormal, Function, sign, avg, cross, curl, split
 )
 from firedrake.fml import Term, keep, drop
 from gusto.configuration import IntegrateByParts, TransportEquationType
@@ -35,6 +35,16 @@ def __init__(self, equation, variable):
         super().__init__(equation, variable, transport)
 
         self.transport_equation_type = self.original_form.terms[0].get(transport)
+        
+        if self.transport_equation_type == TransportEquationType.tracer_conservative:
+            # Extract associated density variable:
+            tracer_idx = self.equation.field_names.index(variable)
+            print(tracer_idx)
+            tracer = self.equation.active_tracers[tracer_idx]
+            density_idx = self.equation.field_names.index(tracer.density_name)
+            print(density_idx)
+            X = self.equation.X
+            self.conservative_density = split(self.equation.X)[density_idx]
 
     def replace_form(self, equation):
         """
@@ -175,17 +185,12 @@ def __init__(self, equation, variable, ibp=IntegrateByParts.ONCE,
             form = upwind_vector_invariant_form(self.domain, self.test, self.field, ibp=ibp)
 
         elif self.transport_equation_type == TransportEquationType.tracer_conservative:
-            # Can use the conservative equation, but ensuring the correct field
-            # for a mixing ratio.
-            if self.field.variable_type == TracerVariableType.density:
-                form = upwind_conservative_form(self.domain, self.test, self.field, ibp=ibp)
-            elif self.field.variable_type == TracerVariableType.mixing_ratio:
-                ref_density_idx = self.eqn.field_names.index(tracer.density_name)
-                ref_density = split(self.X)[ref_density_idx]
-                q = self.field*ref_density
-                form = upwind_conservative_form(self.domain, self.test, q, ibp=ibp)
-        
-            form = upwind_conservative_form(self.domain, self.test, self.field, ibp=ibp)
+            # Can use the conservative equation for the tracer multiplied
+            # by the associated density
+            #ref_density_idx = self.equation.field_names.index(variable.density_name)
+            #ref_density = split(self.X)[self.density_idx]
+            q = self.field*self.conservative_density
+            form = upwind_continuity_form(self.domain, self.test, q, ibp=ibp)
         else:
             raise NotImplementedError('Upwind transport scheme has not been '
                                       + 'implemented for this transport equation type')

From 649d35e581d1e72b65a385fdd98a2d27233f720b Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Wed, 13 Dec 2023 03:32:49 +0000
Subject: [PATCH 03/21] more changes

---
 gusto/common_forms.py      | 11 +++++-----
 gusto/equations.py         | 42 +++++++++++++++++++++++++++++++-------
 gusto/transport_methods.py |  5 ++---
 3 files changed, 43 insertions(+), 15 deletions(-)

diff --git a/gusto/common_forms.py b/gusto/common_forms.py
index d1355a44a..90073fad5 100644
--- a/gusto/common_forms.py
+++ b/gusto/common_forms.py
@@ -263,7 +263,7 @@ def split_continuity_form(equation):
 
     return equation
     
-def tracer_conservative_form(test, q, ubar):
+def tracer_conservative_form(test, q, rho, ubar):
     u"""
     The form corresponding to the continuity transport operator.
 
@@ -272,15 +272,16 @@ def tracer_conservative_form(test, q, ubar):
 
     Args:
         test (:class:`TestFunction`): the test function.
-        q (:class:`ufl.Expr`): the tracer to be transported. If this
-        is a mixing ratio, q = m*rho. If it is a density, q = rho.
+        q (:class:`ufl.Expr`): the tracer to be transported. 
+        rho (:class:`ufl.Expr`): the reference density that is to
+        be multiplied by q before taking the divergence
         ubar (:class:`ufl.Expr`): the transporting velocity.
 
     Returns:
         class:`LabelledForm`: a labelled transport form.
     """
-    
-    L = inner(test, div(outer(q, ubar)))*dx
+    q_rho = q*rho
+    L = inner(test, div(outer(q_rho, ubar)))*dx
     form = transporting_velocity(L, ubar)
     
     return transport(form, TransportEquationType.tracer_conservative)
diff --git a/gusto/equations.py b/gusto/equations.py
index ab39733cd..297fefa5b 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -271,7 +271,29 @@ def generate_mass_terms(self):
 
         for i, (test, field_name) in enumerate(zip(self.tests, self.field_names)):
             prog = split(self.X)[i]
+            
+            # Hack for now
+            # I can't be sure that the index of active tracers is the same,
+            # so I need to search for the active_tracer.name == prog
+            #if self.active_tracers[i].transport_eqn == TransportEquationType.tracer_conservative:
+            #    tracer = self.active_tracers[i]
+            #    print(tracer.name)
+            #    ref_density_idx = self.field_names.index(tracer.density_name)
+            #    ref_density = split(self.X)[ref_density_idx]
+            #    q = prog*ref_density
+            #    mass = subject(prognostic(inner(q, test)*dx, field_name), self.X)
+            #print(self.active_tracers[i])
+            
+            #if self.active_tracers.name
+            
+            #if prog.transport_eqn == TransportEquationType.tracer_conservative:
+            #    print('yo')
+            
+            #else:
+            #    mass = subject(prognostic(inner(prog, test)*dx, field_name), self.X)
+            
             mass = subject(prognostic(inner(prog, test)*dx, field_name), self.X)
+        
             if i == 0:
                 mass_form = time_derivative(mass)
             else:
@@ -430,7 +452,7 @@ def add_tracers_to_prognostics(self, domain, active_tracers):
             else:
                 raise TypeError(f'Tracers must be ActiveTracer objects, not {type(tracer)}')
 
-    def generate_tracer_mass_terms(self):
+    def generate_tracer_mass_terms(self, active_tracers):
         """
         Builds the weak time derivative terms for the equation set.
 
@@ -445,16 +467,20 @@ def generate_tracer_mass_terms(self):
             :class:`LabelledForm`: a labelled form containing the mass terms.
         """
 
-        for _, tracer in enumerate(active_tracers):
+        for i, tracer in enumerate(active_tracers):
             idx = self.field_names.index(tracer.name)
+            #print(idx)
             tracer_prog = split(self.X)[idx]
             tracer_test = self.tests[idx]
             if tracer.transport_eqn == TransportEquationType.tracer_conservative:
                 ref_density_idx = self.field_names.index(tracer.density_name)
                 ref_density = split(self.X)[ref_density_idx]
+                #print(i)
+                #print(ref_density_idx)
                 q = tracer_prog*ref_density
-                mass = subject(prognostic(inner(q, test)*dx, field_name), self.X)
-                
+                mass = subject(prognostic(inner(q, tracer_test)*dx, self.field_name), self.X)
+            else:
+                mass = subject(prognostic(inner(tracer_prog, tracer_test)*dx, self.field_name), self.X)
             if i == 0:
                 mass_form = time_derivative(mass)
             else:
@@ -508,9 +534,9 @@ def generate_tracer_transport_terms(self, active_tracers):
                 elif tracer.transport_eqn == TransportEquationType.tracer_conservative:
                     ref_density_idx = self.field_names.index(tracer.density_name)
                     ref_density = split(self.X)[ref_density_idx]
-                    q = tracer_prog*ref_density
+                    #q = tracer_prog*ref_density
                     tracer_adv = prognostic(
-                        tracer_conservative_form(tracer_test, q, u),
+                        tracer_conservative_form(tracer_test, tracer_prog, ref_density, u),
                         tracer.name)
                     
                 else:
@@ -664,7 +690,9 @@ def __init__(self, domain, active_tracers, Vu=None):
         self.tests = TestFunctions(W)
         self.X = Function(W)
         
-        mass_form = self.generate_tracer_mass_terms()
+        print(full_field_name)
+        
+        mass_form = self.generate_tracer_mass_terms(active_tracers)
 
         self.residual = subject(mass_form, self.X)
 
diff --git a/gusto/transport_methods.py b/gusto/transport_methods.py
index c890bd261..b5569168d 100644
--- a/gusto/transport_methods.py
+++ b/gusto/transport_methods.py
@@ -39,11 +39,10 @@ def __init__(self, equation, variable):
         if self.transport_equation_type == TransportEquationType.tracer_conservative:
             # Extract associated density variable:
             tracer_idx = self.equation.field_names.index(variable)
-            print(tracer_idx)
+            #print(tracer_idx)
             tracer = self.equation.active_tracers[tracer_idx]
             density_idx = self.equation.field_names.index(tracer.density_name)
-            print(density_idx)
-            X = self.equation.X
+            #print(density_idx)
             self.conservative_density = split(self.equation.X)[density_idx]
 
     def replace_form(self, equation):

From f94df7c7a9a5a69d6fdad8e8f36bb24254861041 Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Wed, 13 Dec 2023 21:57:07 +0000
Subject: [PATCH 04/21] DGUpwind scheme for tracer_conservative

---
 gusto/equations.py         | 30 ++++++++++++------
 gusto/transport_methods.py | 64 ++++++++++++++++++++++++++++++++++++--
 2 files changed, 83 insertions(+), 11 deletions(-)

diff --git a/gusto/equations.py b/gusto/equations.py
index 297fefa5b..a21c1a34d 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -271,7 +271,7 @@ def generate_mass_terms(self):
 
         for i, (test, field_name) in enumerate(zip(self.tests, self.field_names)):
             prog = split(self.X)[i]
-            
+            print(i)
             # Hack for now
             # I can't be sure that the index of active tracers is the same,
             # so I need to search for the active_tracer.name == prog
@@ -294,10 +294,14 @@ def generate_mass_terms(self):
             
             mass = subject(prognostic(inner(prog, test)*dx, field_name), self.X)
         
+            print(mass)
+        
             if i == 0:
                 mass_form = time_derivative(mass)
             else:
                 mass_form += time_derivative(mass)
+                
+            print(mass_form)
 
         return mass_form
 
@@ -469,23 +473,32 @@ def generate_tracer_mass_terms(self, active_tracers):
 
         for i, tracer in enumerate(active_tracers):
             idx = self.field_names.index(tracer.name)
-            #print(idx)
+            print(idx)
             tracer_prog = split(self.X)[idx]
             tracer_test = self.tests[idx]
+            print(self.field_names[idx])
             if tracer.transport_eqn == TransportEquationType.tracer_conservative:
                 ref_density_idx = self.field_names.index(tracer.density_name)
                 ref_density = split(self.X)[ref_density_idx]
                 #print(i)
                 #print(ref_density_idx)
                 q = tracer_prog*ref_density
-                mass = subject(prognostic(inner(q, tracer_test)*dx, self.field_name), self.X)
+                mass = subject(prognostic(inner(q, tracer_test)*dx, self.field_names[idx]), self.X)
+                #mass = subject(inner(ref_density, prognostic(inner(tracer_prog, tracer_test), self.field_names[idx]))*dx, self.X)
+                #mass = subject(inner(ref_density, prognostic(inner(tracer_prog, tracer_test), tracer.name))*dx, self.X)
+                #mass = subject(prognostic(inner(tracer_prog, tracer_test)*dx, self.field_names[idx]), self.X)
+                #mass = prognostic(inner(q, tracer_test)*dx, tracer.name)
             else:
-                mass = subject(prognostic(inner(tracer_prog, tracer_test)*dx, self.field_name), self.X)
+                mass = subject(prognostic(inner(tracer_prog, tracer_test)*dx, self.field_names[idx]), self.X)
+                #mass = prognostic(inner(tracer_prog, tracer_test)*dx, tracer.name)
+            print(mass)
             if i == 0:
                 mass_form = time_derivative(mass)
             else:
                 mass_form += time_derivative(mass)
 
+            print(mass_form)
+
         return mass_form
 
     def generate_tracer_transport_terms(self, active_tracers):
@@ -628,9 +641,10 @@ def __init__(self, domain, active_tracers, Vu=None):
         self.tests = TestFunctions(W)
         self.X = Function(W)
 
-        mass_form = self.generate_mass_terms()
+        #mass_form = self.generate_mass_terms()
 
-        self.residual = subject(mass_form, self.X)
+        #self.residual = subject(mass_form, self.X)
+        self.residual = self.generate_mass_terms()
 
         # Add transport of tracers
         self.residual += self.generate_tracer_transport_terms(active_tracers)
@@ -691,10 +705,8 @@ def __init__(self, domain, active_tracers, Vu=None):
         self.X = Function(W)
         
         print(full_field_name)
-        
-        mass_form = self.generate_tracer_mass_terms(active_tracers)
 
-        self.residual = subject(mass_form, self.X)
+        self.residual = self.generate_tracer_mass_terms(active_tracers)
 
         # Add transport of tracers
         self.residual += self.generate_tracer_transport_terms(active_tracers)        
diff --git a/gusto/transport_methods.py b/gusto/transport_methods.py
index b5569168d..9818015b8 100644
--- a/gusto/transport_methods.py
+++ b/gusto/transport_methods.py
@@ -188,8 +188,8 @@ def __init__(self, equation, variable, ibp=IntegrateByParts.ONCE,
             # by the associated density
             #ref_density_idx = self.equation.field_names.index(variable.density_name)
             #ref_density = split(self.X)[self.density_idx]
-            q = self.field*self.conservative_density
-            form = upwind_continuity_form(self.domain, self.test, q, ibp=ibp)
+            #q = self.field*self.conservative_density
+            form = upwind_tracer_conservative_form(self.domain, self.test, self.field, self.conservative_density, ibp=ibp)
         else:
             raise NotImplementedError('Upwind transport scheme has not been '
                                       + 'implemented for this transport equation type')
@@ -314,6 +314,66 @@ def upwind_continuity_form(domain, test, q, ibp=IntegrateByParts.ONCE, outflow=F
     form = transporting_velocity(L, ubar)
 
     return ibp_label(transport(form, TransportEquationType.conservative), ibp)
+    
+def upwind_tracer_conservative_form(domain, test, q, rho, ibp=IntegrateByParts.ONCE, outflow=False):
+    u"""
+    The form corresponding to the DG upwind continuity transport operator.
+
+    This discretises ∇.(u*q*rho), for transporting velocity u, transported
+    variable q, and its reference density, rho. Although the tracer q obeys an advection
+    equation, the transport term is in a conservative form. 
+
+    Args:
+        domain (:class:`Domain`): the model's domain object, containing the
+            mesh and the compatible function spaces.
+        test (:class:`TestFunction`): the test function.
+        q (:class:`ufl.Expr`): the variable to be transported.
+        rho (:class:`ufl.Expr`): the reference density for the tracer.
+        ibp (:class:`IntegrateByParts`, optional): an enumerator representing
+            the number of times to integrate by parts. Defaults to
+            `IntegrateByParts.ONCE`.
+        outflow (bool, optional): whether to include outflow at the domain
+            boundaries, through exterior facet terms. Defaults to False.
+
+    Raises:
+        ValueError: Can only use outflow option when the integration by parts
+            option is not "never".
+
+    Returns:
+        class:`LabelledForm`: a labelled transport form.
+    """
+
+    if outflow and ibp == IntegrateByParts.NEVER:
+        raise ValueError("outflow is True and ibp is None are incompatible options")
+    Vu = domain.spaces("HDiv")
+    dS_ = (dS_v + dS_h) if Vu.extruded else dS
+    ubar = Function(Vu)
+
+    if ibp == IntegrateByParts.ONCE:
+        L = -inner(grad(test), outer(inner(q,rho), ubar))*dx
+    else:
+        L = inner(test, div(outer(inner(q,rho), ubar)))*dx
+
+    if ibp != IntegrateByParts.NEVER:
+        n = FacetNormal(domain.mesh)
+        un = 0.5*(dot(ubar, n) + abs(dot(ubar, n)))
+
+        L += dot(jump(test), (un('+')*q('+')*rho('+') - un('-')*q('-')*rho('-')))*dS_
+
+        if ibp == IntegrateByParts.TWICE:
+            L -= (inner(test('+'), dot(ubar('+'), n('+'))*q('+')*rho('+'))
+                  + inner(test('-'), dot(ubar('-'), n('-'))*q('-')*rho('-')))*dS_
+
+    if outflow:
+        n = FacetNormal(domain.mesh)
+        un = 0.5*(dot(ubar, n) + abs(dot(ubar, n)))
+        L += test*un*q*rho*(ds_v + ds_t + ds_b)
+
+    form = transporting_velocity(L, ubar)
+    
+    print('DGUpwind with tracer conservative, called')
+
+    return ibp_label(transport(form, TransportEquationType.tracer_conservative), ibp)
 
 
 def vector_manifold_advection_form(domain, test, q, ibp=IntegrateByParts.ONCE, outflow=False):

From 9ca039527266483b8df5772d35dae1e420f55942 Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Tue, 19 Dec 2023 02:41:31 +0000
Subject: [PATCH 05/21] tidy up descriptions

---
 gusto/active_tracers.py    |  7 ++---
 gusto/common_forms.py      | 10 +++----
 gusto/configuration.py     |  4 +--
 gusto/equations.py         | 54 ++++++++++++++------------------------
 gusto/transport_methods.py |  8 +-----
 5 files changed, 31 insertions(+), 52 deletions(-)

diff --git a/gusto/active_tracers.py b/gusto/active_tracers.py
index 93d0a1e73..3bc187a94 100644
--- a/gusto/active_tracers.py
+++ b/gusto/active_tracers.py
@@ -63,10 +63,11 @@ def __init__(self, name, space, variable_type,
                 the type of tracer variable (e.g. mixing ratio or density).
             transport_eqn (:class:`TransportEquationType`, optional): enumerator
                 indicating the type of transport equation to be solved (e.g.
-                advective). Defaults to `TransportEquationType.advective`.#
+                advective). Defaults to `TransportEquationType.advective`.
             density_name (str): the name of the associated density for a mixing
-                ratio when using the tracer_conservative transport. 
-                Defaults to None.
+                ratio when using the tracer_conservative transport. Defaults to None,
+                but raises an error if tracer_conservative transport is used without
+                a specified density.
             phase (:class:`Phases`, optional): enumerator indicating the phase
                 of the tracer variable. Defaults to `Phases.gas`.
             chemical (str, optional): string to describe the chemical that this
diff --git a/gusto/common_forms.py b/gusto/common_forms.py
index 90073fad5..6032f4a8a 100644
--- a/gusto/common_forms.py
+++ b/gusto/common_forms.py
@@ -267,14 +267,14 @@ def tracer_conservative_form(test, q, rho, ubar):
     u"""
     The form corresponding to the continuity transport operator.
 
-    This describes ∇.(u*q) for transporting velocity u and a 
-    transported tracer, q.
+    This describes ∇.(u*q*rho) for transporting velocity u and a 
+    transported tracer (mixing ratio), q, with an associated density, rho.
 
     Args:
         test (:class:`TestFunction`): the test function.
-        q (:class:`ufl.Expr`): the tracer to be transported. 
-        rho (:class:`ufl.Expr`): the reference density that is to
-        be multiplied by q before taking the divergence
+        q (:class:`ufl.Expr`): the tracer to be transported.
+        rho (:class:`ufl.Expr`): the reference density that will
+        mulitply with q before taking the divergence.
         ubar (:class:`ufl.Expr`): the transporting velocity.
 
     Returns:
diff --git a/gusto/configuration.py b/gusto/configuration.py
index b2a3acfa0..0cd904509 100644
--- a/gusto/configuration.py
+++ b/gusto/configuration.py
@@ -31,8 +31,8 @@ class TransportEquationType(Enum):
     conservative: ∂q/∂t + ∇.(u*q) = 0                                         \n
     vector_invariant: ∂q/∂t + (∇×q)×u + (1/2)∇(q.u) + (1/2)[(∇q).u -(∇u).q)] = 0
     circulation: ∂q/∂t + (∇×q)×u + non-transport terms = 0
-    tracer_conservative: to fill in. for transporting a tracer that obeys an
-    advection equation in a conservative manner.
+    tracer_conservative: ∂(q*rho)/∂t + ∇.(u*q*rho) = 0, for a reference density of rho
+    for the tracer, q.
     """
 
     no_transport = 702
diff --git a/gusto/equations.py b/gusto/equations.py
index a21c1a34d..709a1fcaa 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -271,31 +271,7 @@ def generate_mass_terms(self):
 
         for i, (test, field_name) in enumerate(zip(self.tests, self.field_names)):
             prog = split(self.X)[i]
-            print(i)
-            # Hack for now
-            # I can't be sure that the index of active tracers is the same,
-            # so I need to search for the active_tracer.name == prog
-            #if self.active_tracers[i].transport_eqn == TransportEquationType.tracer_conservative:
-            #    tracer = self.active_tracers[i]
-            #    print(tracer.name)
-            #    ref_density_idx = self.field_names.index(tracer.density_name)
-            #    ref_density = split(self.X)[ref_density_idx]
-            #    q = prog*ref_density
-            #    mass = subject(prognostic(inner(q, test)*dx, field_name), self.X)
-            #print(self.active_tracers[i])
-            
-            #if self.active_tracers.name
-            
-            #if prog.transport_eqn == TransportEquationType.tracer_conservative:
-            #    print('yo')
-            
-            #else:
-            #    mass = subject(prognostic(inner(prog, test)*dx, field_name), self.X)
-            
             mass = subject(prognostic(inner(prog, test)*dx, field_name), self.X)
-        
-            print(mass)
-        
             if i == 0:
                 mass_form = time_derivative(mass)
             else:
@@ -465,6 +441,7 @@ def generate_tracer_mass_terms(self, active_tracers):
         
         The mass terms can differ depending on the tracer type. A mixing ratio
         that is being transported conservatively will need its mass form multiplied
+        by a reference density.
         
 
         Returns:
@@ -480,18 +457,23 @@ def generate_tracer_mass_terms(self, active_tracers):
             if tracer.transport_eqn == TransportEquationType.tracer_conservative:
                 ref_density_idx = self.field_names.index(tracer.density_name)
                 ref_density = split(self.X)[ref_density_idx]
-                #print(i)
-                #print(ref_density_idx)
                 q = tracer_prog*ref_density
                 mass = subject(prognostic(inner(q, tracer_test)*dx, self.field_names[idx]), self.X)
-                #mass = subject(inner(ref_density, prognostic(inner(tracer_prog, tracer_test), self.field_names[idx]))*dx, self.X)
-                #mass = subject(inner(ref_density, prognostic(inner(tracer_prog, tracer_test), tracer.name))*dx, self.X)
-                #mass = subject(prognostic(inner(tracer_prog, tracer_test)*dx, self.field_names[idx]), self.X)
-                #mass = prognostic(inner(q, tracer_test)*dx, tracer.name)
+                #term1 = ref_density*time_derivative(inner(tracer_prog, tracer_test))
+                #term2 = tracer_prog*time_derivative(inner(ref_density, tracer_test))
+                #mass_1 = subject(prognostic(term1*dx, self.field_names[idx]), self.X)
+                #mass_2 = subject(prognostic(term2*dx, self.field_names[idx]), self.X)
+                #if i == 0:
+                #    mass_form = mass_1
+                #    mass_form += mass_2
+                #else:
+                #    mass_form += mass_1
+                #    mass_form += mass_2
+
             else:
                 mass = subject(prognostic(inner(tracer_prog, tracer_test)*dx, self.field_names[idx]), self.X)
                 #mass = prognostic(inner(tracer_prog, tracer_test)*dx, tracer.name)
-            print(mass)
+            
             if i == 0:
                 mass_form = time_derivative(mass)
             else:
@@ -641,9 +623,6 @@ def __init__(self, domain, active_tracers, Vu=None):
         self.tests = TestFunctions(W)
         self.X = Function(W)
 
-        #mass_form = self.generate_mass_terms()
-
-        #self.residual = subject(mass_form, self.X)
         self.residual = self.generate_mass_terms()
 
         # Add transport of tracers
@@ -709,7 +688,12 @@ def __init__(self, domain, active_tracers, Vu=None):
         self.residual = self.generate_tracer_mass_terms(active_tracers)
 
         # Add transport of tracers
-        self.residual += self.generate_tracer_transport_terms(active_tracers)        
+        self.residual += self.generate_tracer_transport_terms(active_tracers)      
+        
+        #self.residual = subject(self.generate_tracer_mass_terms(active_tracers),self.X)
+
+        # Add transport of tracers
+        #self.residual += subject(self.generate_tracer_transport_terms(active_tracers),self.X)         
 
 
 
diff --git a/gusto/transport_methods.py b/gusto/transport_methods.py
index 9818015b8..a4424acda 100644
--- a/gusto/transport_methods.py
+++ b/gusto/transport_methods.py
@@ -38,11 +38,10 @@ def __init__(self, equation, variable):
         
         if self.transport_equation_type == TransportEquationType.tracer_conservative:
             # Extract associated density variable:
+            # This does assume the same ordering of tracers and fields...
             tracer_idx = self.equation.field_names.index(variable)
-            #print(tracer_idx)
             tracer = self.equation.active_tracers[tracer_idx]
             density_idx = self.equation.field_names.index(tracer.density_name)
-            #print(density_idx)
             self.conservative_density = split(self.equation.X)[density_idx]
 
     def replace_form(self, equation):
@@ -184,11 +183,6 @@ def __init__(self, equation, variable, ibp=IntegrateByParts.ONCE,
             form = upwind_vector_invariant_form(self.domain, self.test, self.field, ibp=ibp)
 
         elif self.transport_equation_type == TransportEquationType.tracer_conservative:
-            # Can use the conservative equation for the tracer multiplied
-            # by the associated density
-            #ref_density_idx = self.equation.field_names.index(variable.density_name)
-            #ref_density = split(self.X)[self.density_idx]
-            #q = self.field*self.conservative_density
             form = upwind_tracer_conservative_form(self.domain, self.test, self.field, self.conservative_density, ibp=ibp)
         else:
             raise NotImplementedError('Upwind transport scheme has not been '

From b14f4a07c6ff79aaa93abdcad66eb22e6c618289 Mon Sep 17 00:00:00 2001
From: Tom Bendall <thomas.bendall@metoffice.gov.uk>
Date: Thu, 21 Dec 2023 18:15:49 +0000
Subject: [PATCH 06/21] add mass weighted label to time derivative and
 debugging statements. Almost working (but we don't have consistency!)

---
 gusto/equations.py           | 26 +++++++-------
 gusto/labels.py              |  1 +
 gusto/time_discretisation.py | 68 ++++++++++++++++++++++++++++++++++--
 3 files changed, 80 insertions(+), 15 deletions(-)

diff --git a/gusto/equations.py b/gusto/equations.py
index 709a1fcaa..ee3f0b530 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -14,7 +14,7 @@
 from gusto.fields import PrescribedFields
 from gusto.labels import (
     time_derivative, transport, prognostic, hydrostatic, linearisation,
-    pressure_gradient, coriolis
+    pressure_gradient, coriolis, mass_weighted
 )
 from gusto.thermodynamics import exner_pressure
 from gusto.common_forms import (
@@ -276,7 +276,7 @@ def generate_mass_terms(self):
                 mass_form = time_derivative(mass)
             else:
                 mass_form += time_derivative(mass)
-                
+
             print(mass_form)
 
         return mass_form
@@ -438,11 +438,11 @@ def generate_tracer_mass_terms(self, active_tracers):
 
         Generates the weak time derivative terms ("mass terms") for all the
         prognostic variables of the equation set.
-        
+
         The mass terms can differ depending on the tracer type. A mixing ratio
         that is being transported conservatively will need its mass form multiplied
         by a reference density.
-        
+
 
         Returns:
             :class:`LabelledForm`: a labelled form containing the mass terms.
@@ -458,7 +458,9 @@ def generate_tracer_mass_terms(self, active_tracers):
                 ref_density_idx = self.field_names.index(tracer.density_name)
                 ref_density = split(self.X)[ref_density_idx]
                 q = tracer_prog*ref_density
-                mass = subject(prognostic(inner(q, tracer_test)*dx, self.field_names[idx]), self.X)
+                mass = mass_weighted(
+                    subject(prognostic(inner(q, tracer_test)*dx, self.field_names[idx]),
+                            self.X), (ref_density_idx, inner(tracer_prog, tracer_test)*dx))
                 #term1 = ref_density*time_derivative(inner(tracer_prog, tracer_test))
                 #term2 = tracer_prog*time_derivative(inner(ref_density, tracer_test))
                 #mass_1 = subject(prognostic(term1*dx, self.field_names[idx]), self.X)
@@ -473,7 +475,7 @@ def generate_tracer_mass_terms(self, active_tracers):
             else:
                 mass = subject(prognostic(inner(tracer_prog, tracer_test)*dx, self.field_names[idx]), self.X)
                 #mass = prognostic(inner(tracer_prog, tracer_test)*dx, tracer.name)
-            
+
             if i == 0:
                 mass_form = time_derivative(mass)
             else:
@@ -533,7 +535,7 @@ def generate_tracer_transport_terms(self, active_tracers):
                     tracer_adv = prognostic(
                         tracer_conservative_form(tracer_test, tracer_prog, ref_density, u),
                         tracer.name)
-                    
+
                 else:
                     raise ValueError(f'Transport eqn {tracer.transport_eqn} not recognised')
 
@@ -627,7 +629,7 @@ def __init__(self, domain, active_tracers, Vu=None):
 
         # Add transport of tracers
         self.residual += self.generate_tracer_transport_terms(active_tracers)
-        
+
 class ConservativeCoupledTransportEquation(PrognosticEquationSet):
     u"""
     Discretises the transport equation,               \n
@@ -682,18 +684,18 @@ def __init__(self, domain, active_tracers, Vu=None):
 
         self.tests = TestFunctions(W)
         self.X = Function(W)
-        
+
         print(full_field_name)
 
         self.residual = self.generate_tracer_mass_terms(active_tracers)
 
         # Add transport of tracers
-        self.residual += self.generate_tracer_transport_terms(active_tracers)      
-        
+        self.residual += self.generate_tracer_transport_terms(active_tracers)
+
         #self.residual = subject(self.generate_tracer_mass_terms(active_tracers),self.X)
 
         # Add transport of tracers
-        #self.residual += subject(self.generate_tracer_transport_terms(active_tracers),self.X)         
+        #self.residual += subject(self.generate_tracer_transport_terms(active_tracers),self.X)
 
 
 
diff --git a/gusto/labels.py b/gusto/labels.py
index ec0fb4af4..8bdb1eb9e 100644
--- a/gusto/labels.py
+++ b/gusto/labels.py
@@ -90,6 +90,7 @@ def __call__(self, target, value=None):
 # ---------------------------------------------------------------------------- #
 
 time_derivative = Label("time_derivative")
+mass_weighted = Label("mass_weighted", validator=lambda value: type(value) == tuple)
 implicit = Label("implicit")
 explicit = Label("explicit")
 transport = Label("transport", validator=lambda value: type(value) == TransportEquationType)
diff --git a/gusto/time_discretisation.py b/gusto/time_discretisation.py
index 79c8af94e..8419e3987 100644
--- a/gusto/time_discretisation.py
+++ b/gusto/time_discretisation.py
@@ -14,14 +14,14 @@
     NonlinearVariationalSolver, DirichletBC, split, Constant
 )
 from firedrake.fml import (
-    replace_subject, replace_test_function, Term, all_terms, drop
+    replace_subject, replace_test_function, Term, all_terms, drop, keep
 )
 from firedrake.formmanipulation import split_form
 from firedrake.utils import cached_property
 
 from gusto.configuration import EmbeddedDGOptions, RecoveryOptions
 from gusto.labels import (time_derivative, prognostic, physics_label,
-                          implicit, explicit)
+                          implicit, explicit, mass_weighted)
 from gusto.logging import logger, DEBUG, logging_ksp_monitor_true_residual
 from gusto.wrappers import *
 
@@ -519,6 +519,17 @@ def setup(self, equation, apply_bcs=True, *active_labels):
         self.x0 = Function(self.fs)
         self.x1 = Function(self.fs)
 
+        # Work out if any sub-functions need special mass weighted handling
+        self.mass_weighted_idxs = []
+        self.density_idxs = []
+        for t in self.residual.terms:
+            if t.has_label(mass_weighted):
+                # Get the index in the mixed function space of this term,
+                # from the prognostic field name
+                idx = self.equation.field_names.index(t.get(prognostic))
+                self.mass_weighted_idxs.append(idx)
+                self.density_idxs.append(t.get(mass_weighted)[0])
+
     @cached_property
     def lhs(self):
         """Set up the discretisation's left hand side (the time derivative)."""
@@ -807,7 +818,23 @@ def setup(self, equation, apply_bcs=True, *active_labels):
     @cached_property
     def lhs(self):
         """Set up the discretisation's left hand side (the time derivative)."""
-        return super(ExplicitMultistage, self).lhs
+
+        # Special handling for mass weighted time derivatives
+        # Explicit multi-stage solves for "k", which should be linear increment
+        # in the prognostic variables -- so we need to drop the mass weighted
+        # part to make the LHS linear. The non-mass weighted time derivative is
+        # (for now) stored in the mass weighted label
+        self.residual = self.residual.label_map(
+            lambda t: t.has_label(mass_weighted),
+            map_if_true=lambda t: Term(t.get(mass_weighted)[1], t.labels),
+            map_if_false=keep)
+
+        l = self.residual.label_map(
+            lambda t: t.has_label(time_derivative),
+            map_if_true=replace_subject(self.x_out, self.idx),
+            map_if_false=drop)
+
+        return l.form
 
     @cached_property
     def rhs(self):
@@ -835,8 +862,19 @@ def rhs(self):
 
         return r.form
 
+    def print_stats(self, mixed_field, message):
+        min1 = np.min(mixed_field.dat[0].data)
+        max1 = np.max(mixed_field.dat[0].data)
+        min2 = np.min(mixed_field.dat[1].data)
+        max2 = np.max(mixed_field.dat[1].data)
+        logger.info(f'{message}: {min1} {max1} {min2} {max2}')
+
     def solve_stage(self, x0, stage):
         self.x1.assign(x0)
+
+        # Use x0 as a first guess (otherwise may not converge)
+        self.x_out.assign(x0)
+
         for i in range(stage):
             self.x1.assign(self.x1 + self.dt*self.butcher_matrix[stage-1, i]*self.k[i])
         for evaluate in self.evaluate_source:
@@ -844,6 +882,28 @@ def solve_stage(self, x0, stage):
         if self.limiter is not None:
             self.limiter.apply(self.x1)
         self.solver.solve()
+
+        self.print_stats(self.x1, f'x1 {stage}')
+
+        self.print_stats(self.x_out, 'x_out before')
+
+        # Need to divide by the updated density field if the LHS has a mass
+        # weighted time derivative
+        if len(self.mass_weighted_idxs) > 0:
+            # Loop through mixing ratios and the corresponding densities
+            for rho_idx, m_idx in zip(self.density_idxs, self.mass_weighted_idxs):
+                # Calculate updated density
+                rho = Function(self.fs[rho_idx])
+                rho.assign(x0.subfunctions[rho_idx])
+                for i in range(stage):
+                    rho.assign(rho + self.dt*self.butcher_matrix[stage, i]*self.k[i].subfunctions[rho_idx])
+                rho.assign(rho + self.dt*self.butcher_matrix[stage, stage]*self.x_out.subfunctions[rho_idx])
+                logger.info(f'rho updated: {np.min(rho.dat.data)} {np.max(rho.dat.data)}')
+                # Divide mixing ratio increment by updated density
+                k_m = self.x_out.subfunctions[m_idx]
+                k_m.interpolate(k_m / rho)
+        self.print_stats(self.x_out, 'x_out_after')
+
         self.k[stage].assign(self.x_out)
 
         if (stage == self.nStages - 1):
@@ -855,6 +915,8 @@ def solve_stage(self, x0, stage):
             if self.limiter is not None:
                 self.limiter.apply(self.x1)
 
+            self.print_stats(self.x1, f'x1 end')
+
     def apply_cycle(self, x_out, x_in):
         """
         Apply the time discretisation through a single sub-step.

From f3064c3b376388b62e8cbe0bfff438add35808fd Mon Sep 17 00:00:00 2001
From: Tom Bendall <thomas.bendall@metoffice.gov.uk>
Date: Sun, 21 Jan 2024 15:38:49 +0000
Subject: [PATCH 07/21] implement new way of writing explicit RK schemes

---
 gusto/equations.py                            |  36 +--
 gusto/labels.py                               |   2 +-
 gusto/time_discretisation.py                  | 262 +++++++++++-------
 gusto/transport_methods.py                    |   8 +-
 .../model/test_time_discretisation.py         |  12 +-
 5 files changed, 181 insertions(+), 139 deletions(-)

diff --git a/gusto/equations.py b/gusto/equations.py
index ee3f0b530..8b64baf6a 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -14,7 +14,7 @@
 from gusto.fields import PrescribedFields
 from gusto.labels import (
     time_derivative, transport, prognostic, hydrostatic, linearisation,
-    pressure_gradient, coriolis, mass_weighted
+    pressure_gradient, coriolis
 )
 from gusto.thermodynamics import exner_pressure
 from gusto.common_forms import (
@@ -277,8 +277,6 @@ def generate_mass_terms(self):
             else:
                 mass_form += time_derivative(mass)
 
-            print(mass_form)
-
         return mass_form
 
     # ======================================================================== #
@@ -450,39 +448,24 @@ def generate_tracer_mass_terms(self, active_tracers):
 
         for i, tracer in enumerate(active_tracers):
             idx = self.field_names.index(tracer.name)
-            print(idx)
             tracer_prog = split(self.X)[idx]
             tracer_test = self.tests[idx]
-            print(self.field_names[idx])
+
             if tracer.transport_eqn == TransportEquationType.tracer_conservative:
                 ref_density_idx = self.field_names.index(tracer.density_name)
                 ref_density = split(self.X)[ref_density_idx]
                 q = tracer_prog*ref_density
-                mass = mass_weighted(
-                    subject(prognostic(inner(q, tracer_test)*dx, self.field_names[idx]),
-                            self.X), (ref_density_idx, inner(tracer_prog, tracer_test)*dx))
-                #term1 = ref_density*time_derivative(inner(tracer_prog, tracer_test))
-                #term2 = tracer_prog*time_derivative(inner(ref_density, tracer_test))
-                #mass_1 = subject(prognostic(term1*dx, self.field_names[idx]), self.X)
-                #mass_2 = subject(prognostic(term2*dx, self.field_names[idx]), self.X)
-                #if i == 0:
-                #    mass_form = mass_1
-                #    mass_form += mass_2
-                #else:
-                #    mass_form += mass_1
-                #    mass_form += mass_2
-
+                mass = subject(prognostic(inner(q, tracer_test)*dx,
+                                          self.field_names[idx]), self.X)
             else:
-                mass = subject(prognostic(inner(tracer_prog, tracer_test)*dx, self.field_names[idx]), self.X)
-                #mass = prognostic(inner(tracer_prog, tracer_test)*dx, tracer.name)
+                mass = subject(prognostic(inner(tracer_prog, tracer_test)*dx,
+                                          self.field_names[idx]), self.X)
 
             if i == 0:
                 mass_form = time_derivative(mass)
             else:
                 mass_form += time_derivative(mass)
 
-            print(mass_form)
-
         return mass_form
 
     def generate_tracer_transport_terms(self, active_tracers):
@@ -685,18 +668,11 @@ def __init__(self, domain, active_tracers, Vu=None):
         self.tests = TestFunctions(W)
         self.X = Function(W)
 
-        print(full_field_name)
-
         self.residual = self.generate_tracer_mass_terms(active_tracers)
 
         # Add transport of tracers
         self.residual += self.generate_tracer_transport_terms(active_tracers)
 
-        #self.residual = subject(self.generate_tracer_mass_terms(active_tracers),self.X)
-
-        # Add transport of tracers
-        #self.residual += subject(self.generate_tracer_transport_terms(active_tracers),self.X)
-
 
 
 # ============================================================================ #
diff --git a/gusto/labels.py b/gusto/labels.py
index 8bdb1eb9e..129184056 100644
--- a/gusto/labels.py
+++ b/gusto/labels.py
@@ -90,7 +90,7 @@ def __call__(self, target, value=None):
 # ---------------------------------------------------------------------------- #
 
 time_derivative = Label("time_derivative")
-mass_weighted = Label("mass_weighted", validator=lambda value: type(value) == tuple)
+mass_weighted = Label("mass_weighted", validator=lambda value: type(value) == str)
 implicit = Label("implicit")
 explicit = Label("explicit")
 transport = Label("transport", validator=lambda value: type(value) == TransportEquationType)
diff --git a/gusto/time_discretisation.py b/gusto/time_discretisation.py
index 8419e3987..c9c25cf06 100644
--- a/gusto/time_discretisation.py
+++ b/gusto/time_discretisation.py
@@ -21,7 +21,7 @@
 
 from gusto.configuration import EmbeddedDGOptions, RecoveryOptions
 from gusto.labels import (time_derivative, prognostic, physics_label,
-                          implicit, explicit, mass_weighted)
+                          implicit, explicit)
 from gusto.logging import logger, DEBUG, logging_ksp_monitor_true_residual
 from gusto.wrappers import *
 
@@ -519,17 +519,6 @@ def setup(self, equation, apply_bcs=True, *active_labels):
         self.x0 = Function(self.fs)
         self.x1 = Function(self.fs)
 
-        # Work out if any sub-functions need special mass weighted handling
-        self.mass_weighted_idxs = []
-        self.density_idxs = []
-        for t in self.residual.terms:
-            if t.has_label(mass_weighted):
-                # Get the index in the mixed function space of this term,
-                # from the prognostic field name
-                idx = self.equation.field_names.index(t.get(prognostic))
-                self.mass_weighted_idxs.append(idx)
-                self.density_idxs.append(t.get(mass_weighted)[0])
-
     @cached_property
     def lhs(self):
         """Set up the discretisation's left hand side (the time derivative)."""
@@ -761,8 +750,9 @@ class ExplicitMultistage(ExplicitTimeDiscretisation):
     #  [ b_0  b_1  .       b_s]
     # ---------------------------------------------------------------------------
 
-    def __init__(self, domain, butcher_matrix, field_name=None, fixed_subcycles=None,
-                 subcycle_by_courant=None, solver_parameters=None,
+    def __init__(self, domain, butcher_matrix, field_name=None,
+                 fixed_subcycles=None, subcycle_by_courant=None,
+                 increment_form=True, solver_parameters=None,
                  limiter=None, options=None):
         """
         Args:
@@ -781,6 +771,9 @@ def __init__(self, domain, butcher_matrix, field_name=None, fixed_subcycles=None
                 for one sub-cycle. Defaults to None, in which case adaptive
                 sub-cycling is not used. This option cannot be specified with the
                 `fixed_subcycles` argument.
+            increment_form (bool, optional): whether to write the RK scheme in
+                "increment form", solving for increments rather than updated
+                fields. Defaults to True.
             solver_parameters (dict, optional): dictionary of parameters to
                 pass to the underlying solver. Defaults to None.
             limiter (:class:`Limiter` object, optional): a limiter to apply to
@@ -797,6 +790,7 @@ def __init__(self, domain, butcher_matrix, field_name=None, fixed_subcycles=None
                          limiter=limiter, options=options)
         self.butcher_matrix = butcher_matrix
         self.nbutcher = int(np.shape(self.butcher_matrix)[0])
+        self.increment_form = increment_form
 
     @property
     def nStages(self):
@@ -813,109 +807,162 @@ def setup(self, equation, apply_bcs=True, *active_labels):
         """
         super().setup(equation, apply_bcs, *active_labels)
 
-        self.k = [Function(self.fs) for i in range(self.nStages)]
+        if not self.increment_form:
+            self.field_i = [Function(self.fs) for i in range(self.nStages+1)]
+        else:
+            self.k = [Function(self.fs) for i in range(self.nStages)]
+
+    @cached_property
+    def solver(self):
+        if self.increment_form:
+            return super().solver
+        else:
+            # In this case, don't set snes_type to ksp only, as we do want the
+            # outer Newton iteration
+            solver_list = []
+
+            for stage in range(self.nStages):
+                # setup linear solver using lhs and rhs defined in derived class
+                problem = NonlinearVariationalProblem(self.lhs[stage].form - self.rhs[stage].form, self.field_i[stage+1], bcs=self.bcs)
+                solver_name = self.field_name+self.__class__.__name__+str(stage)
+                solver =  NonlinearVariationalSolver(problem, solver_parameters=self.solver_parameters,
+                                                     options_prefix=solver_name)
+                solver_list.append(solver)
+            return solver_list
+
 
     @cached_property
     def lhs(self):
         """Set up the discretisation's left hand side (the time derivative)."""
 
-        # Special handling for mass weighted time derivatives
-        # Explicit multi-stage solves for "k", which should be linear increment
-        # in the prognostic variables -- so we need to drop the mass weighted
-        # part to make the LHS linear. The non-mass weighted time derivative is
-        # (for now) stored in the mass weighted label
-        self.residual = self.residual.label_map(
-            lambda t: t.has_label(mass_weighted),
-            map_if_true=lambda t: Term(t.get(mass_weighted)[1], t.labels),
-            map_if_false=keep)
+        if self.increment_form:
+            l = self.residual.label_map(
+                lambda t: t.has_label(time_derivative),
+                map_if_true=replace_subject(self.x_out, self.idx),
+                map_if_false=drop)
 
-        l = self.residual.label_map(
-            lambda t: t.has_label(time_derivative),
-            map_if_true=replace_subject(self.x_out, self.idx),
-            map_if_false=drop)
+            return l.form
+
+        else:
+            lhs_list = []
+            for stage in range(self.nStages):
+                l = self.residual.label_map(
+                    lambda t: t.has_label(time_derivative),
+                    map_if_true=replace_subject(self.field_i[stage+1], self.idx),
+                    map_if_false=drop)
+                lhs_list.append(l)
+
+            return lhs_list
 
-        return l.form
 
     @cached_property
     def rhs(self):
         """Set up the time discretisation's right hand side."""
-        r = self.residual.label_map(
-            all_terms,
-            map_if_true=replace_subject(self.x1, old_idx=self.idx))
 
-        r = r.label_map(
-            lambda t: t.has_label(time_derivative),
-            map_if_true=drop,
-            map_if_false=lambda t: -1*t)
-
-        # If there are no active labels, we may have no terms at this point
-        # So that we can still do xnp1 = xn, put in a zero term here
-        if len(r.terms) == 0:
-            logger.warning('No terms detected for RHS of explicit problem. '
-                           + 'Adding a zero term to avoid failure.')
-            null_term = Constant(0.0)*self.residual.label_map(
+        if self.increment_form:
+            r = self.residual.label_map(
+                all_terms,
+                map_if_true=replace_subject(self.x1, old_idx=self.idx))
+
+            r = r.label_map(
                 lambda t: t.has_label(time_derivative),
-                # Drop label from this
-                map_if_true=lambda t: time_derivative.remove(t),
-                map_if_false=drop)
-            r += null_term
+                map_if_true=drop,
+                map_if_false=lambda t: -1*t)
+
+            # If there are no active labels, we may have no terms at this point
+            # So that we can still do xnp1 = xn, put in a zero term here
+            if self.increment_form and len(r.terms) == 0:
+                logger.warning('No terms detected for RHS of explicit problem. '
+                            + 'Adding a zero term to avoid failure.')
+                null_term = Constant(0.0)*self.residual.label_map(
+                    lambda t: t.has_label(time_derivative),
+                    # Drop label from this
+                    map_if_true=lambda t: time_derivative.remove(t),
+                    map_if_false=drop)
+                r += null_term
+
+            return r.form
 
-        return r.form
+        else:
+            rhs_list = []
 
-    def print_stats(self, mixed_field, message):
-        min1 = np.min(mixed_field.dat[0].data)
-        max1 = np.max(mixed_field.dat[0].data)
-        min2 = np.min(mixed_field.dat[1].data)
-        max2 = np.max(mixed_field.dat[1].data)
-        logger.info(f'{message}: {min1} {max1} {min2} {max2}')
+            for stage in range(self.nStages):
+                r = self.residual.label_map(
+                    lambda t: t.has_label(time_derivative),
+                    map_if_true=replace_subject(self.field_i[0], old_idx=self.idx),
+                    map_if_false=replace_subject(self.field_i[0], old_idx=self.idx))
 
-    def solve_stage(self, x0, stage):
-        self.x1.assign(x0)
+                r = r.label_map(
+                    lambda t: t.has_label(time_derivative),
+                    map_if_true=keep,
+                    map_if_false=lambda t: -self.butcher_matrix[stage, 0]*self.dt*t)
 
-        # Use x0 as a first guess (otherwise may not converge)
-        self.x_out.assign(x0)
+                for i in range(1, stage+1):
+                    r_i = self.residual.label_map(
+                        lambda t: t.has_label(time_derivative),
+                        map_if_true=drop,
+                        map_if_false=replace_subject(self.field_i[i], old_idx=self.idx)
+                    )
 
-        for i in range(stage):
-            self.x1.assign(self.x1 + self.dt*self.butcher_matrix[stage-1, i]*self.k[i])
-        for evaluate in self.evaluate_source:
-            evaluate(self.x1, self.dt)
-        if self.limiter is not None:
-            self.limiter.apply(self.x1)
-        self.solver.solve()
+                    r -= self.butcher_matrix[stage, i]*self.dt*r_i
 
-        self.print_stats(self.x1, f'x1 {stage}')
-
-        self.print_stats(self.x_out, 'x_out before')
-
-        # Need to divide by the updated density field if the LHS has a mass
-        # weighted time derivative
-        if len(self.mass_weighted_idxs) > 0:
-            # Loop through mixing ratios and the corresponding densities
-            for rho_idx, m_idx in zip(self.density_idxs, self.mass_weighted_idxs):
-                # Calculate updated density
-                rho = Function(self.fs[rho_idx])
-                rho.assign(x0.subfunctions[rho_idx])
-                for i in range(stage):
-                    rho.assign(rho + self.dt*self.butcher_matrix[stage, i]*self.k[i].subfunctions[rho_idx])
-                rho.assign(rho + self.dt*self.butcher_matrix[stage, stage]*self.x_out.subfunctions[rho_idx])
-                logger.info(f'rho updated: {np.min(rho.dat.data)} {np.max(rho.dat.data)}')
-                # Divide mixing ratio increment by updated density
-                k_m = self.x_out.subfunctions[m_idx]
-                k_m.interpolate(k_m / rho)
-        self.print_stats(self.x_out, 'x_out_after')
+                rhs_list.append(r)
 
-        self.k[stage].assign(self.x_out)
+            return rhs_list
+
+    def solve_stage(self, x0, stage):
 
-        if (stage == self.nStages - 1):
+        if self.increment_form:
             self.x1.assign(x0)
-            for i in range(self.nStages):
-                self.x1.assign(self.x1 + self.dt*self.butcher_matrix[stage, i]*self.k[i])
-            self.x1.assign(self.x1)
 
+            # Use x0 as a first guess (otherwise may not converge)
+            self.x_out.assign(x0)
+
+            for i in range(stage):
+                self.x1.assign(self.x1 + self.dt*self.butcher_matrix[stage-1, i]*self.k[i])
+            for evaluate in self.evaluate_source:
+                evaluate(self.x1, self.dt)
             if self.limiter is not None:
                 self.limiter.apply(self.x1)
+            self.solver.solve()
+
+            self.k[stage].assign(self.x_out)
+
+            if (stage == self.nStages - 1):
+                self.x1.assign(x0)
+                for i in range(self.nStages):
+                    self.x1.assign(self.x1 + self.dt*self.butcher_matrix[stage, i]*self.k[i])
+                self.x1.assign(self.x1)
+
+                if self.limiter is not None:
+                    self.limiter.apply(self.x1)
+
+        else:
+            # Set initial field
+            if stage == 0:
+                self.field_i[0].assign(x0)
+
+            # Use x0 as a first guess (otherwise may not converge)
+            self.field_i[stage+1].assign(x0)
+
+            # Update field_i for physics / limiters
+            for evaluate in self.evaluate_source:
+                # TODO: not implemented! Here we need to evaluate the m-th term
+                # in the i-th RHS with field_m
+                raise NotImplementedError(
+                    'Physics not implemented with RK schemes that do not use '
+                    + 'the increment form')
+            if self.limiter is not None:
+                self.limiter.apply(self.field_i[stage])
+
+            # Obtain field_ip1 = field_n - dt* sum_m{c_im*F[field_m]}
+            self.solver[stage].solve()
+
+            if (stage == self.nStages - 1):
+                self.x1.assign(self.field_i[stage+1])
+                if self.limiter is not None:
+                    self.limiter.apply(self.x1)
 
-            self.print_stats(self.x1, f'x1 end')
 
     def apply_cycle(self, x_out, x_in):
         """
@@ -944,8 +991,8 @@ class ForwardEuler(ExplicitMultistage):
     y^(n+1) = y^n + dt*k0                                                       \n
     """
     def __init__(self, domain, field_name=None, fixed_subcycles=None,
-                 subcycle_by_courant=None, solver_parameters=None,
-                 limiter=None, options=None):
+                 subcycle_by_courant=None, increment_form=True,
+                 solver_parameters=None, limiter=None, options=None):
         """
         Args:
             domain (:class:`Domain`): the model's domain object, containing the
@@ -961,6 +1008,9 @@ def __init__(self, domain, field_name=None, fixed_subcycles=None,
                 for one sub-cycle. Defaults to None, in which case adaptive
                 sub-cycling is not used. This option cannot be specified with the
                 `fixed_subcycles` argument.
+            increment_form (bool, optional): whether to write the RK scheme in
+                "increment form", solving for increments rather than updated
+                fields. Defaults to True.
             solver_parameters (dict, optional): dictionary of parameters to
                 pass to the underlying solver. Defaults to None.
             limiter (:class:`Limiter` object, optional): a limiter to apply to
@@ -974,6 +1024,7 @@ def __init__(self, domain, field_name=None, fixed_subcycles=None,
         super().__init__(domain, butcher_matrix, field_name=field_name,
                          fixed_subcycles=fixed_subcycles,
                          subcycle_by_courant=subcycle_by_courant,
+                         increment_form=increment_form,
                          solver_parameters=solver_parameters,
                          limiter=limiter, options=options)
 
@@ -989,8 +1040,8 @@ class SSPRK3(ExplicitMultistage):
     y^(n+1) = y^n + (1/6)*dt*(k0 + k1 + 4*k2)                                 \n
     """
     def __init__(self, domain, field_name=None, fixed_subcycles=None,
-                 subcycle_by_courant=None, solver_parameters=None,
-                 limiter=None, options=None):
+                 subcycle_by_courant=None, increment_form=True,
+                 solver_parameters=None, limiter=None, options=None):
         """
         Args:
             domain (:class:`Domain`): the model's domain object, containing the
@@ -1006,6 +1057,9 @@ def __init__(self, domain, field_name=None, fixed_subcycles=None,
                 for one sub-cycle. Defaults to None, in which case adaptive
                 sub-cycling is not used. This option cannot be specified with the
                 `fixed_subcycles` argument.
+            increment_form (bool, optional): whether to write the RK scheme in
+                "increment form", solving for increments rather than updated
+                fields. Defaults to True.
             solver_parameters (dict, optional): dictionary of parameters to
                 pass to the underlying solver. Defaults to None.
             limiter (:class:`Limiter` object, optional): a limiter to apply to
@@ -1016,9 +1070,11 @@ def __init__(self, domain, field_name=None, fixed_subcycles=None,
                 recovery method. Defaults to None.
         """
         butcher_matrix = np.array([[1., 0., 0.], [1./4., 1./4., 0.], [1./6., 1./6., 2./3.]])
+
         super().__init__(domain, butcher_matrix, field_name=field_name,
                          fixed_subcycles=fixed_subcycles,
                          subcycle_by_courant=subcycle_by_courant,
+                         increment_form=increment_form,
                          solver_parameters=solver_parameters,
                          limiter=limiter, options=options)
 
@@ -1039,7 +1095,8 @@ class RK4(ExplicitMultistage):
     where superscripts indicate the time-level.                               \n
     """
     def __init__(self, domain, field_name=None, fixed_subcycles=None,
-                 subcycle_by_courant=None, solver_parameters=None,
+                 subcycle_by_courant=None, increment_form=True,
+                 solver_parameters=None,
                  limiter=None, options=None):
         """
         Args:
@@ -1056,6 +1113,9 @@ def __init__(self, domain, field_name=None, fixed_subcycles=None,
                 for one sub-cycle. Defaults to None, in which case adaptive
                 sub-cycling is not used. This option cannot be specified with the
                 `fixed_subcycles` argument.
+            increment_form (bool, optional): whether to write the RK scheme in
+                "increment form", solving for increments rather than updated
+                fields. Defaults to True.
             solver_parameters (dict, optional): dictionary of parameters to
                 pass to the underlying solver. Defaults to None.
             limiter (:class:`Limiter` object, optional): a limiter to apply to
@@ -1069,6 +1129,7 @@ def __init__(self, domain, field_name=None, fixed_subcycles=None,
         super().__init__(domain, butcher_matrix, field_name=field_name,
                          fixed_subcycles=fixed_subcycles,
                          subcycle_by_courant=subcycle_by_courant,
+                         increment_form=increment_form,
                          solver_parameters=solver_parameters,
                          limiter=limiter, options=options)
 
@@ -1087,8 +1148,8 @@ class Heun(ExplicitMultistage):
     number.
     """
     def __init__(self, domain, field_name=None, fixed_subcycles=None,
-                 subcycle_by_courant=None, solver_parameters=None,
-                 limiter=None, options=None):
+                 subcycle_by_courant=None, increment_form=True,
+                 solver_parameters=None, limiter=None, options=None):
         """
         Args:
             domain (:class:`Domain`): the model's domain object, containing the
@@ -1104,6 +1165,9 @@ def __init__(self, domain, field_name=None, fixed_subcycles=None,
                 for one sub-cycle. Defaults to None, in which case adaptive
                 sub-cycling is not used. This option cannot be specified with the
                 `fixed_subcycles` argument.
+            increment_form (bool, optional): whether to write the RK scheme in
+                "increment form", solving for increments rather than updated
+                fields. Defaults to True.
             solver_parameters (dict, optional): dictionary of parameters to
                 pass to the underlying solver. Defaults to None.
             limiter (:class:`Limiter` object, optional): a limiter to apply to
@@ -1117,6 +1181,7 @@ def __init__(self, domain, field_name=None, fixed_subcycles=None,
         super().__init__(domain, butcher_matrix, field_name=field_name,
                          fixed_subcycles=fixed_subcycles,
                          subcycle_by_courant=subcycle_by_courant,
+                         increment_form=increment_form,
                          solver_parameters=solver_parameters,
                          limiter=limiter, options=options)
 
@@ -1919,7 +1984,6 @@ def apply(self, x_out, *x_in):
         """
         if self.initial_timesteps < self.nlevels-1:
             self.initial_timesteps += 1
-            print(self.initial_timesteps)
             solver = self.solver0
         else:
             solver = self.solver
diff --git a/gusto/transport_methods.py b/gusto/transport_methods.py
index a4424acda..274ba326b 100644
--- a/gusto/transport_methods.py
+++ b/gusto/transport_methods.py
@@ -35,7 +35,7 @@ def __init__(self, equation, variable):
         super().__init__(equation, variable, transport)
 
         self.transport_equation_type = self.original_form.terms[0].get(transport)
-        
+
         if self.transport_equation_type == TransportEquationType.tracer_conservative:
             # Extract associated density variable:
             # This does assume the same ordering of tracers and fields...
@@ -308,14 +308,14 @@ def upwind_continuity_form(domain, test, q, ibp=IntegrateByParts.ONCE, outflow=F
     form = transporting_velocity(L, ubar)
 
     return ibp_label(transport(form, TransportEquationType.conservative), ibp)
-    
+
 def upwind_tracer_conservative_form(domain, test, q, rho, ibp=IntegrateByParts.ONCE, outflow=False):
     u"""
     The form corresponding to the DG upwind continuity transport operator.
 
     This discretises ∇.(u*q*rho), for transporting velocity u, transported
     variable q, and its reference density, rho. Although the tracer q obeys an advection
-    equation, the transport term is in a conservative form. 
+    equation, the transport term is in a conservative form.
 
     Args:
         domain (:class:`Domain`): the model's domain object, containing the
@@ -364,8 +364,6 @@ def upwind_tracer_conservative_form(domain, test, q, rho, ibp=IntegrateByParts.O
         L += test*un*q*rho*(ds_v + ds_t + ds_b)
 
     form = transporting_velocity(L, ubar)
-    
-    print('DGUpwind with tracer conservative, called')
 
     return ibp_label(transport(form, TransportEquationType.tracer_conservative), ibp)
 
diff --git a/integration-tests/model/test_time_discretisation.py b/integration-tests/model/test_time_discretisation.py
index 6f114fe90..f4d9ddf2f 100644
--- a/integration-tests/model/test_time_discretisation.py
+++ b/integration-tests/model/test_time_discretisation.py
@@ -8,8 +8,10 @@ def run(timestepper, tmax, f_end):
     return norm(timestepper.fields("f") - f_end) / norm(f_end)
 
 
-@pytest.mark.parametrize("scheme", ["ssprk", "TrapeziumRule", "ImplicitMidpoint", "QinZhang",
-                                    "RK4", "Heun", "BDF2", "TR_BDF2", "AdamsBashforth", "Leapfrog", "AdamsMoulton"])
+@pytest.mark.parametrize("scheme",
+    ["ssprk3_increment", "TrapeziumRule", "ImplicitMidpoint", "QinZhang", "RK4",
+     "Heun", "BDF2", "TR_BDF2", "AdamsBashforth", "Leapfrog", "AdamsMoulton",
+     "ssprk3_predictor"])
 def test_time_discretisation(tmpdir, scheme, tracer_setup):
     if (scheme == "AdamsBashforth"):
         # Tighter stability constraints
@@ -25,8 +27,10 @@ def test_time_discretisation(tmpdir, scheme, tracer_setup):
         V = domain.spaces("DG")
         eqn = AdvectionEquation(domain, V, "f")
 
-    if scheme == "ssprk":
-        transport_scheme = SSPRK3(domain)
+    if scheme == "ssprk3_increment":
+        transport_scheme = SSPRK3(domain, increment_form=True)
+    elif scheme == "ssprk3_predictor":
+        transport_scheme = SSPRK3(domain, increment_form=False)
     elif scheme == "TrapeziumRule":
         transport_scheme = TrapeziumRule(domain)
     elif scheme == "ImplicitMidpoint":

From ef4582ff4cf910223e8fa7a70864189033ce72b3 Mon Sep 17 00:00:00 2001
From: Tom Bendall <thomas.bendall@metoffice.gov.uk>
Date: Sun, 21 Jan 2024 15:50:23 +0000
Subject: [PATCH 08/21] remove mass weighted label

---
 gusto/labels.py | 1 -
 1 file changed, 1 deletion(-)

diff --git a/gusto/labels.py b/gusto/labels.py
index 129184056..ec0fb4af4 100644
--- a/gusto/labels.py
+++ b/gusto/labels.py
@@ -90,7 +90,6 @@ def __call__(self, target, value=None):
 # ---------------------------------------------------------------------------- #
 
 time_derivative = Label("time_derivative")
-mass_weighted = Label("mass_weighted", validator=lambda value: type(value) == str)
 implicit = Label("implicit")
 explicit = Label("explicit")
 transport = Label("transport", validator=lambda value: type(value) == TransportEquationType)

From 738b9513738edeaecfef42421d5fd0394f68f1c8 Mon Sep 17 00:00:00 2001
From: Tom Bendall <thomas.bendall@metoffice.gov.uk>
Date: Sun, 21 Jan 2024 16:02:59 +0000
Subject: [PATCH 09/21] lint fixes

---
 gusto/active_tracers.py                       | 11 +++++---
 gusto/common_forms.py                         |  9 ++++---
 gusto/equations.py                            | 26 ++++++++-----------
 gusto/time_discretisation.py                  | 14 +++++-----
 gusto/transport_methods.py                    |  8 +++---
 .../model/test_time_discretisation.py         |  8 +++---
 6 files changed, 39 insertions(+), 37 deletions(-)

diff --git a/gusto/active_tracers.py b/gusto/active_tracers.py
index 3bc187a94..60329e5fb 100644
--- a/gusto/active_tracers.py
+++ b/gusto/active_tracers.py
@@ -85,12 +85,15 @@ def __init__(self, name, space, variable_type,
         self.phase = phase
         self.chemical = chemical
 
-        if (variable_type == TracerVariableType.density and transport_eqn == TransportEquationType.advective):
+        if (variable_type == TracerVariableType.density
+                and transport_eqn == TransportEquationType.advective):
             logger.warning('Active tracer initialised which describes a '
                            + 'density but solving the advective transport eqn')
-                           
-        if (transport_eqn == TransportEquationType.tracer_conservative and density_name == None):
-            raise ValueError(f'Active tracer {name} using tracer conservative transport needs an associated density.')
+
+        if (transport_eqn == TransportEquationType.tracer_conservative
+                and density_name is None):
+            raise ValueError(f'Active tracer {name} using tracer conservative '
+                             + 'transport needs an associated density.')
 
 
 class WaterVapour(ActiveTracer):
diff --git a/gusto/common_forms.py b/gusto/common_forms.py
index 6032f4a8a..83dad21ff 100644
--- a/gusto/common_forms.py
+++ b/gusto/common_forms.py
@@ -262,12 +262,13 @@ def split_continuity_form(equation):
                 map_if_true=drop)
 
     return equation
-    
+
+
 def tracer_conservative_form(test, q, rho, ubar):
     u"""
     The form corresponding to the continuity transport operator.
 
-    This describes ∇.(u*q*rho) for transporting velocity u and a 
+    This describes ∇.(u*q*rho) for transporting velocity u and a
     transported tracer (mixing ratio), q, with an associated density, rho.
 
     Args:
@@ -280,9 +281,9 @@ def tracer_conservative_form(test, q, rho, ubar):
     Returns:
         class:`LabelledForm`: a labelled transport form.
     """
+
     q_rho = q*rho
     L = inner(test, div(outer(q_rho, ubar)))*dx
     form = transporting_velocity(L, ubar)
-    
+
     return transport(form, TransportEquationType.tracer_conservative)
-    
diff --git a/gusto/equations.py b/gusto/equations.py
index 8b64baf6a..7fa0eb071 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -514,10 +514,9 @@ def generate_tracer_transport_terms(self, active_tracers):
                 elif tracer.transport_eqn == TransportEquationType.tracer_conservative:
                     ref_density_idx = self.field_names.index(tracer.density_name)
                     ref_density = split(self.X)[ref_density_idx]
-                    #q = tracer_prog*ref_density
                     tracer_adv = prognostic(
-                        tracer_conservative_form(tracer_test, tracer_prog, ref_density, u),
-                        tracer.name)
+                        tracer_conservative_form(tracer_test, tracer_prog,
+                                                 ref_density, u), tracer.name)
 
                 else:
                     raise ValueError(f'Transport eqn {tracer.transport_eqn} not recognised')
@@ -613,18 +612,17 @@ def __init__(self, domain, active_tracers, Vu=None):
         # Add transport of tracers
         self.residual += self.generate_tracer_transport_terms(active_tracers)
 
+
 class ConservativeCoupledTransportEquation(PrognosticEquationSet):
     u"""
-    Discretises the transport equation,               \n
-    ∂q/∂t + (u.∇)q = F,
-    with the application of active tracers.
-    As there are multiple tracers or species that are
-    interacting, q and F are vectors.
-    This equation can be enhanced through the addition of
-    sources or sinks (F) by applying it with physics schemes.
-    This takes in tracers that might obey different forms
-    of the transport equation (i.e. advective, conservative)
-    but will evolve all the fields in a conservative manner.
+    Discretises the transport equation,                                       \n
+    ∂q/∂t + (u.∇)q = F,                                                       \n
+    with the application of active tracers. As there are multiple tracers or
+    species that are interacting, q and F are vectors. This equation can be
+    enhanced through the addition of sources or sinks (F) by applying it with
+    physics schemes. This takes in tracers that might obey different forms of
+    the transport equation (i.e. advective, conservative) but will evolve all
+    the fields in a conservative manner.
     """
     def __init__(self, domain, active_tracers, Vu=None):
         """
@@ -674,12 +672,10 @@ def __init__(self, domain, active_tracers, Vu=None):
         self.residual += self.generate_tracer_transport_terms(active_tracers)
 
 
-
 # ============================================================================ #
 # Specified Equation Sets
 # ============================================================================ #
 
-
 class ShallowWaterEquations(PrognosticEquationSet):
     u"""
     Class for the (rotating) shallow-water equations, which evolve the velocity
diff --git a/gusto/time_discretisation.py b/gusto/time_discretisation.py
index c9c25cf06..e93c2226d 100644
--- a/gusto/time_discretisation.py
+++ b/gusto/time_discretisation.py
@@ -823,14 +823,16 @@ def solver(self):
 
             for stage in range(self.nStages):
                 # setup linear solver using lhs and rhs defined in derived class
-                problem = NonlinearVariationalProblem(self.lhs[stage].form - self.rhs[stage].form, self.field_i[stage+1], bcs=self.bcs)
+                problem = NonlinearVariationalProblem(
+                    self.lhs[stage].form - self.rhs[stage].form,
+                    self.field_i[stage+1], bcs=self.bcs)
                 solver_name = self.field_name+self.__class__.__name__+str(stage)
-                solver =  NonlinearVariationalSolver(problem, solver_parameters=self.solver_parameters,
-                                                     options_prefix=solver_name)
+                solver = NonlinearVariationalSolver(
+                    problem, solver_parameters=self.solver_parameters,
+                    options_prefix=solver_name)
                 solver_list.append(solver)
             return solver_list
 
-
     @cached_property
     def lhs(self):
         """Set up the discretisation's left hand side (the time derivative)."""
@@ -854,7 +856,6 @@ def lhs(self):
 
             return lhs_list
 
-
     @cached_property
     def rhs(self):
         """Set up the time discretisation's right hand side."""
@@ -873,7 +874,7 @@ def rhs(self):
             # So that we can still do xnp1 = xn, put in a zero term here
             if self.increment_form and len(r.terms) == 0:
                 logger.warning('No terms detected for RHS of explicit problem. '
-                            + 'Adding a zero term to avoid failure.')
+                               + 'Adding a zero term to avoid failure.')
                 null_term = Constant(0.0)*self.residual.label_map(
                     lambda t: t.has_label(time_derivative),
                     # Drop label from this
@@ -963,7 +964,6 @@ def solve_stage(self, x0, stage):
                 if self.limiter is not None:
                     self.limiter.apply(self.x1)
 
-
     def apply_cycle(self, x_out, x_in):
         """
         Apply the time discretisation through a single sub-step.
diff --git a/gusto/transport_methods.py b/gusto/transport_methods.py
index 274ba326b..cd43b921f 100644
--- a/gusto/transport_methods.py
+++ b/gusto/transport_methods.py
@@ -309,7 +309,9 @@ def upwind_continuity_form(domain, test, q, ibp=IntegrateByParts.ONCE, outflow=F
 
     return ibp_label(transport(form, TransportEquationType.conservative), ibp)
 
-def upwind_tracer_conservative_form(domain, test, q, rho, ibp=IntegrateByParts.ONCE, outflow=False):
+
+def upwind_tracer_conservative_form(domain, test, q, rho,
+                                    ibp=IntegrateByParts.ONCE, outflow=False):
     u"""
     The form corresponding to the DG upwind continuity transport operator.
 
@@ -344,9 +346,9 @@ def upwind_tracer_conservative_form(domain, test, q, rho, ibp=IntegrateByParts.O
     ubar = Function(Vu)
 
     if ibp == IntegrateByParts.ONCE:
-        L = -inner(grad(test), outer(inner(q,rho), ubar))*dx
+        L = -inner(grad(test), outer(inner(q, rho), ubar))*dx
     else:
-        L = inner(test, div(outer(inner(q,rho), ubar)))*dx
+        L = inner(test, div(outer(inner(q, rho), ubar)))*dx
 
     if ibp != IntegrateByParts.NEVER:
         n = FacetNormal(domain.mesh)
diff --git a/integration-tests/model/test_time_discretisation.py b/integration-tests/model/test_time_discretisation.py
index f4d9ddf2f..9523ef75e 100644
--- a/integration-tests/model/test_time_discretisation.py
+++ b/integration-tests/model/test_time_discretisation.py
@@ -8,10 +8,10 @@ def run(timestepper, tmax, f_end):
     return norm(timestepper.fields("f") - f_end) / norm(f_end)
 
 
-@pytest.mark.parametrize("scheme",
-    ["ssprk3_increment", "TrapeziumRule", "ImplicitMidpoint", "QinZhang", "RK4",
-     "Heun", "BDF2", "TR_BDF2", "AdamsBashforth", "Leapfrog", "AdamsMoulton",
-     "ssprk3_predictor"])
+@pytest.mark.parametrize(
+    "scheme", ["ssprk3_increment", "TrapeziumRule", "ImplicitMidpoint",
+               "QinZhang", "RK4", "Heun", "BDF2", "TR_BDF2", "AdamsBashforth",
+               "Leapfrog", "AdamsMoulton", "AdamsMoulton", "ssprk3_predictor"])
 def test_time_discretisation(tmpdir, scheme, tracer_setup):
     if (scheme == "AdamsBashforth"):
         # Tighter stability constraints

From 94643230d10689e1fa27ee05f3002af4cf9a95bf Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Thu, 14 Mar 2024 18:28:18 +0000
Subject: [PATCH 10/21] added a ConservativeCoupledTransportEquation test

---
 .../test_conservative_coupled_transport.py    | 67 +++++++++++++++++++
 unit-tests/test_active_tracer.py              |  3 +-
 2 files changed, 69 insertions(+), 1 deletion(-)
 create mode 100644 integration-tests/equations/test_conservative_coupled_transport.py

diff --git a/integration-tests/equations/test_conservative_coupled_transport.py b/integration-tests/equations/test_conservative_coupled_transport.py
new file mode 100644
index 000000000..df10cd73c
--- /dev/null
+++ b/integration-tests/equations/test_conservative_coupled_transport.py
@@ -0,0 +1,67 @@
+"""
+Tests the CoupledTransportEquation class.
+Two tracers are transported using combinations of advective and
+conservative forms. The tracers are set to be a mixing ratio when
+using the advective form and a density for the conservative form.
+
+"""
+
+from firedrake import norm, BrokenElement
+from gusto import *
+import pytest
+
+
+def run(timestepper, tmax, f_end):
+    timestepper.run(0, tmax)
+    norm1 = norm(timestepper.fields("rho_d") - f_end) / norm(f_end)
+    norm2 = norm(timestepper.fields("m_X") - f_end) / norm(f_end)
+    return norm1, norm2
+
+@pytest.mark.parametrize("m_X_space", ['DG', 'theta'])
+def test_coupled_transport_scalar(tmpdir, m_X_space, tracer_setup):
+
+    setup = tracer_setup(tmpdir, "slice")
+    domain = setup.domain
+    mesh = domain.mesh
+
+    rho_d = ActiveTracer(name='rho_d', space='DG',
+                         variable_type=TracerVariableType.density,
+                         transport_eqn=TransportEquationType.conservative)
+
+    m_X = ActiveTracer(name='m_X', space=m_X_space,
+                     variable_type=TracerVariableType.mixing_ratio,
+                     transport_eqn=TransportEquationType.tracer_conservative,
+                     density_name='rho_d')
+
+    tracers = [rho_d, m_X]
+
+    V = domain.spaces("HDiv")
+    eqn = ConservativeCoupledTransportEquation(domain, active_tracers=tracers, Vu=V)
+
+    if m_X_space == 'theta':
+        V_m_X = domain.spaces(m_X_space)
+        Vt_brok = FunctionSpace(mesh, BrokenElement(V_m_X.ufl_element()))
+        suboptions = {'rho_d':RecoveryOptions(embedding_space=Vt_brok,
+                                          recovered_space=V_m_X,
+                                          project_low_method='recover'),
+                      'm_X': EmbeddedDGOptions()}
+        opts = MixedFSOptions(suboptions=suboptions)
+
+        transport_scheme = SSPRK3(domain, options = opts, increment_form=False)
+
+    transport_scheme = SSPRK3(domain, increment_form=False)
+
+    transport_method = [DGUpwind(eqn, 'rho_d'), DGUpwind(eqn, 'm_X')]
+
+    timestepper = PrescribedTransport(eqn, transport_scheme, setup.io, transport_method)
+
+    # Initial conditions
+    timestepper.fields("rho_d").interpolate(setup.f_init)
+    timestepper.fields("m_X").interpolate(setup.f_init)
+    timestepper.fields("u").project(setup.uexpr)
+
+    error1, error2 = run(timestepper, setup.tmax, setup.f_end)
+    assert error1 < setup.tol, \
+        'The transport error for rho_d is greater than the permitted tolerance'
+    assert error2 < setup.tol, \
+        'The transport error for m_X is greater than the permitted tolerance'
diff --git a/unit-tests/test_active_tracer.py b/unit-tests/test_active_tracer.py
index 8c7907c43..65c8ff56b 100644
--- a/unit-tests/test_active_tracer.py
+++ b/unit-tests/test_active_tracer.py
@@ -11,7 +11,8 @@ def test_tracer_classes():
     names = ['mr_v', 'big_blob']
     spaces = ['V', 'U']
     transport_eqns = [TransportEquationType.advective,
-                      TransportEquationType.no_transport]
+                      TransportEquationType.no_transport,
+                      TransportEquationType.tracer_conservative]
     variable_types = [TracerVariableType.mixing_ratio]
 
     for name, space, transport_eqn in zip(names, spaces, transport_eqns):

From 0525eb1e16af092960ad783d631eeefd19126362 Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Thu, 14 Mar 2024 18:30:22 +0000
Subject: [PATCH 11/21] lint

---
 .../test_conservative_coupled_transport.py        | 15 ++++++++-------
 1 file changed, 8 insertions(+), 7 deletions(-)

diff --git a/integration-tests/equations/test_conservative_coupled_transport.py b/integration-tests/equations/test_conservative_coupled_transport.py
index df10cd73c..6ab46fd25 100644
--- a/integration-tests/equations/test_conservative_coupled_transport.py
+++ b/integration-tests/equations/test_conservative_coupled_transport.py
@@ -17,6 +17,7 @@ def run(timestepper, tmax, f_end):
     norm2 = norm(timestepper.fields("m_X") - f_end) / norm(f_end)
     return norm1, norm2
 
+
 @pytest.mark.parametrize("m_X_space", ['DG', 'theta'])
 def test_coupled_transport_scalar(tmpdir, m_X_space, tracer_setup):
 
@@ -29,9 +30,9 @@ def test_coupled_transport_scalar(tmpdir, m_X_space, tracer_setup):
                          transport_eqn=TransportEquationType.conservative)
 
     m_X = ActiveTracer(name='m_X', space=m_X_space,
-                     variable_type=TracerVariableType.mixing_ratio,
-                     transport_eqn=TransportEquationType.tracer_conservative,
-                     density_name='rho_d')
+                       variable_type=TracerVariableType.mixing_ratio,
+                       transport_eqn=TransportEquationType.tracer_conservative,
+                       density_name='rho_d')
 
     tracers = [rho_d, m_X]
 
@@ -41,13 +42,13 @@ def test_coupled_transport_scalar(tmpdir, m_X_space, tracer_setup):
     if m_X_space == 'theta':
         V_m_X = domain.spaces(m_X_space)
         Vt_brok = FunctionSpace(mesh, BrokenElement(V_m_X.ufl_element()))
-        suboptions = {'rho_d':RecoveryOptions(embedding_space=Vt_brok,
-                                          recovered_space=V_m_X,
-                                          project_low_method='recover'),
+        suboptions = {'rho_d': RecoveryOptions(embedding_space=Vt_brok,
+                                               recovered_space=V_m_X,
+                                               project_low_method='recover'),
                       'm_X': EmbeddedDGOptions()}
         opts = MixedFSOptions(suboptions=suboptions)
 
-        transport_scheme = SSPRK3(domain, options = opts, increment_form=False)
+        transport_scheme = SSPRK3(domain, options=opts, increment_form=False)
 
     transport_scheme = SSPRK3(domain, increment_form=False)
 

From 2819c7a47d6ccbce37235a8094dde12cf06252ab Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Fri, 15 Mar 2024 09:21:37 +0000
Subject: [PATCH 12/21] integrate ConservativeCoupledTransportEquation into
 CoupledTransportEquation

---
 gusto/equations.py                            | 64 +----------------
 .../test_conservative_coupled_transport.py    | 68 -------------------
 .../equations/test_coupled_transport.py       | 52 +++++++++++++-
 3 files changed, 54 insertions(+), 130 deletions(-)
 delete mode 100644 integration-tests/equations/test_conservative_coupled_transport.py

diff --git a/gusto/equations.py b/gusto/equations.py
index c7b9bfbf0..fc7db41a7 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -607,71 +607,13 @@ def __init__(self, domain, active_tracers, Vu=None):
         self.tests = TestFunctions(W)
         self.X = Function(W)
 
-        self.residual = self.generate_mass_terms()
-
-        # Add transport of tracers
-        self.residual += self.generate_tracer_transport_terms(active_tracers)
-
-
-class ConservativeCoupledTransportEquation(PrognosticEquationSet):
-    u"""
-    Discretises the transport equation,                                       \n
-    ∂q/∂t + (u.∇)q = F,                                                       \n
-    with the application of active tracers. As there are multiple tracers or
-    species that are interacting, q and F are vectors. This equation can be
-    enhanced through the addition of sources or sinks (F) by applying it with
-    physics schemes. This takes in tracers that might obey different forms of
-    the transport equation (i.e. advective, conservative) but will evolve all
-    the fields in a conservative manner.
-    """
-    def __init__(self, domain, active_tracers, Vu=None):
-        """
-        Args:
-            domain (:class:`Domain`): the model's domain object, containing the
-                mesh and the compatible function spaces.
-            active_tracers (list): a list of `ActiveTracer` objects
-                that encode the metadata for any active tracers to be included
-                in the equations. This is required for using this class; if there
-                is only a field to be advected, use the AdvectionEquation
-                instead.
-            Vu (:class:`FunctionSpace`, optional): the function space for the
-                velocity field. If this is not specified, uses the HDiv spaces
-                set up by the domain. Defaults to None.
-        """
-
-        self.active_tracers = active_tracers
-        self.terms_to_linearise = {}
-        self.field_names = []
-        self.space_names = {}
-
-        # Build finite element spaces
-        self.spaces = []
-
-        # Add active tracers to the list of prognostics
-        if active_tracers is None:
-            active_tracers = []
-        self.add_tracers_to_prognostics(domain, active_tracers)
-
-        # Make the full mixed function space
-        W = MixedFunctionSpace(self.spaces)
-
-        full_field_name = "_".join(self.field_names)
-        PrognosticEquation.__init__(self, domain, W, full_field_name)
-
-        if Vu is not None:
-            domain.spaces.add_space("HDiv", Vu, overwrite_space=True)
-        V = domain.spaces("HDiv")
-        _ = self.prescribed_fields("u", V)
-
-        self.tests = TestFunctions(W)
-        self.X = Function(W)
-
+        # Add mass forms for the tracers, which will use
+        # mass*density for any tracer_conservative terms
         self.residual = self.generate_tracer_mass_terms(active_tracers)
-
+        
         # Add transport of tracers
         self.residual += self.generate_tracer_transport_terms(active_tracers)
 
-
 # ============================================================================ #
 # Specified Equation Sets
 # ============================================================================ #
diff --git a/integration-tests/equations/test_conservative_coupled_transport.py b/integration-tests/equations/test_conservative_coupled_transport.py
deleted file mode 100644
index 6ab46fd25..000000000
--- a/integration-tests/equations/test_conservative_coupled_transport.py
+++ /dev/null
@@ -1,68 +0,0 @@
-"""
-Tests the CoupledTransportEquation class.
-Two tracers are transported using combinations of advective and
-conservative forms. The tracers are set to be a mixing ratio when
-using the advective form and a density for the conservative form.
-
-"""
-
-from firedrake import norm, BrokenElement
-from gusto import *
-import pytest
-
-
-def run(timestepper, tmax, f_end):
-    timestepper.run(0, tmax)
-    norm1 = norm(timestepper.fields("rho_d") - f_end) / norm(f_end)
-    norm2 = norm(timestepper.fields("m_X") - f_end) / norm(f_end)
-    return norm1, norm2
-
-
-@pytest.mark.parametrize("m_X_space", ['DG', 'theta'])
-def test_coupled_transport_scalar(tmpdir, m_X_space, tracer_setup):
-
-    setup = tracer_setup(tmpdir, "slice")
-    domain = setup.domain
-    mesh = domain.mesh
-
-    rho_d = ActiveTracer(name='rho_d', space='DG',
-                         variable_type=TracerVariableType.density,
-                         transport_eqn=TransportEquationType.conservative)
-
-    m_X = ActiveTracer(name='m_X', space=m_X_space,
-                       variable_type=TracerVariableType.mixing_ratio,
-                       transport_eqn=TransportEquationType.tracer_conservative,
-                       density_name='rho_d')
-
-    tracers = [rho_d, m_X]
-
-    V = domain.spaces("HDiv")
-    eqn = ConservativeCoupledTransportEquation(domain, active_tracers=tracers, Vu=V)
-
-    if m_X_space == 'theta':
-        V_m_X = domain.spaces(m_X_space)
-        Vt_brok = FunctionSpace(mesh, BrokenElement(V_m_X.ufl_element()))
-        suboptions = {'rho_d': RecoveryOptions(embedding_space=Vt_brok,
-                                               recovered_space=V_m_X,
-                                               project_low_method='recover'),
-                      'm_X': EmbeddedDGOptions()}
-        opts = MixedFSOptions(suboptions=suboptions)
-
-        transport_scheme = SSPRK3(domain, options=opts, increment_form=False)
-
-    transport_scheme = SSPRK3(domain, increment_form=False)
-
-    transport_method = [DGUpwind(eqn, 'rho_d'), DGUpwind(eqn, 'm_X')]
-
-    timestepper = PrescribedTransport(eqn, transport_scheme, setup.io, transport_method)
-
-    # Initial conditions
-    timestepper.fields("rho_d").interpolate(setup.f_init)
-    timestepper.fields("m_X").interpolate(setup.f_init)
-    timestepper.fields("u").project(setup.uexpr)
-
-    error1, error2 = run(timestepper, setup.tmax, setup.f_end)
-    assert error1 < setup.tol, \
-        'The transport error for rho_d is greater than the permitted tolerance'
-    assert error2 < setup.tol, \
-        'The transport error for m_X is greater than the permitted tolerance'
diff --git a/integration-tests/equations/test_coupled_transport.py b/integration-tests/equations/test_coupled_transport.py
index e22bbd81d..add9cc35a 100644
--- a/integration-tests/equations/test_coupled_transport.py
+++ b/integration-tests/equations/test_coupled_transport.py
@@ -6,7 +6,7 @@
 
 """
 
-from firedrake import norm
+from firedrake import norm, BrokenElement
 from gusto import *
 import pytest
 
@@ -63,3 +63,53 @@ def test_coupled_transport_scalar(tmpdir, geometry, equation_form1, equation_for
         'The transport error for tracer 1 is greater than the permitted tolerance'
     assert error2 < setup.tol, \
         'The transport error for tracer 2 is greater than the permitted tolerance'
+
+
+@pytest.mark.parametrize("m_X_space", ['DG', 'theta'])
+def test_conservative_coupled_transport(tmpdir, m_X_space, tracer_setup):
+
+    setup = tracer_setup(tmpdir, "slice")
+    domain = setup.domain
+    mesh = domain.mesh
+
+    rho_d = ActiveTracer(name='f1', space='DG',
+                         variable_type=TracerVariableType.density,
+                         transport_eqn=TransportEquationType.conservative)
+
+    m_X = ActiveTracer(name='f2', space=m_X_space,
+                       variable_type=TracerVariableType.mixing_ratio,
+                       transport_eqn=TransportEquationType.tracer_conservative,
+                       density_name='f1')
+
+    tracers = [rho_d, m_X]
+
+    V = domain.spaces("HDiv")
+    eqn = CoupledTransportEquation(domain, active_tracers=tracers, Vu=V)
+
+    if m_X_space == 'theta':
+        V_m_X = domain.spaces(m_X_space)
+        Vt_brok = FunctionSpace(mesh, BrokenElement(V_m_X.ufl_element()))
+        suboptions = {'f1': RecoveryOptions(embedding_space=Vt_brok,
+                                               recovered_space=V_m_X,
+                                               project_low_method='recover'),
+                      'f2': EmbeddedDGOptions()}
+        opts = MixedFSOptions(suboptions=suboptions)
+
+        transport_scheme = SSPRK3(domain, options=opts, increment_form=False)
+
+    transport_scheme = SSPRK3(domain, increment_form=False)
+
+    transport_method = [DGUpwind(eqn, 'f1'), DGUpwind(eqn, 'f2')]
+
+    timestepper = PrescribedTransport(eqn, transport_scheme, setup.io, transport_method)
+
+    # Initial conditions
+    timestepper.fields("f1").interpolate(setup.f_init)
+    timestepper.fields("f2").interpolate(setup.f_init)
+    timestepper.fields("u").project(setup.uexpr)
+
+    error1, error2 = run(timestepper, setup.tmax, setup.f_end)
+    assert error1 < setup.tol, \
+        'The transport error for rho_d is greater than the permitted tolerance'
+    assert error2 < setup.tol, \
+        'The transport error for m_X is greater than the permitted tolerance'
\ No newline at end of file

From 6b695b21ab342a36bf324b6fd44d5483ccc31f9e Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Fri, 15 Mar 2024 09:26:47 +0000
Subject: [PATCH 13/21] lint

---
 gusto/equations.py | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/gusto/equations.py b/gusto/equations.py
index 15baf9f18..5e95af6b1 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -376,8 +376,7 @@ def set_no_normal_flow_bcs(self, domain, no_normal_flow_bc_ids):
                 + 'when there is a variable called "u" and none was found')
 
         Vu = domain.spaces("HDiv")
-        # we only apply no normal-flow BCs when extruded mesh is non periodic
-        if Vu.extruded and not Vu.ufl_domain().topology.extruded_periodic:
+        if Vu.extruded:
             self.bcs['u'].append(DirichletBC(Vu, 0.0, "bottom"))
             self.bcs['u'].append(DirichletBC(Vu, 0.0, "top"))
         for id in no_normal_flow_bc_ids:
@@ -611,10 +610,11 @@ def __init__(self, domain, active_tracers, Vu=None):
         # Add mass forms for the tracers, which will use
         # mass*density for any tracer_conservative terms
         self.residual = self.generate_tracer_mass_terms(active_tracers)
-        
+
         # Add transport of tracers
         self.residual += self.generate_tracer_transport_terms(active_tracers)
 
+
 # ============================================================================ #
 # Specified Equation Sets
 # ============================================================================ #

From f6c886ab914fc6c089cbdae227aafd204ec7a29d Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Fri, 15 Mar 2024 09:29:18 +0000
Subject: [PATCH 14/21] lint

---
 integration-tests/equations/test_coupled_transport.py | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/integration-tests/equations/test_coupled_transport.py b/integration-tests/equations/test_coupled_transport.py
index add9cc35a..b314a3236 100644
--- a/integration-tests/equations/test_coupled_transport.py
+++ b/integration-tests/equations/test_coupled_transport.py
@@ -90,8 +90,8 @@ def test_conservative_coupled_transport(tmpdir, m_X_space, tracer_setup):
         V_m_X = domain.spaces(m_X_space)
         Vt_brok = FunctionSpace(mesh, BrokenElement(V_m_X.ufl_element()))
         suboptions = {'f1': RecoveryOptions(embedding_space=Vt_brok,
-                                               recovered_space=V_m_X,
-                                               project_low_method='recover'),
+                                            recovered_space=V_m_X,
+                                            project_low_method='recover'),
                       'f2': EmbeddedDGOptions()}
         opts = MixedFSOptions(suboptions=suboptions)
 
@@ -112,4 +112,4 @@ def test_conservative_coupled_transport(tmpdir, m_X_space, tracer_setup):
     assert error1 < setup.tol, \
         'The transport error for rho_d is greater than the permitted tolerance'
     assert error2 < setup.tol, \
-        'The transport error for m_X is greater than the permitted tolerance'
\ No newline at end of file
+        'The transport error for m_X is greater than the permitted tolerance'

From 301c7ffb7006e97a0a07005bd265f0b709f8069e Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Fri, 15 Mar 2024 09:33:57 +0000
Subject: [PATCH 15/21] Update comments for the generate_tracer_mass_terms
 routine

---
 gusto/equations.py | 17 ++++++++---------
 1 file changed, 8 insertions(+), 9 deletions(-)

diff --git a/gusto/equations.py b/gusto/equations.py
index 5e95af6b1..e0c67f140 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -432,18 +432,17 @@ def add_tracers_to_prognostics(self, domain, active_tracers):
 
     def generate_tracer_mass_terms(self, active_tracers):
         """
-        Builds the weak time derivative terms for the equation set.
-
-        Generates the weak time derivative terms ("mass terms") for all the
-        prognostic variables of the equation set.
-
-        The mass terms can differ depending on the tracer type. A mixing ratio
-        that is being transported conservatively will need its mass form multiplied
-        by a reference density.
+        Adds the mass forms for the active tracers to the equation set.
 
+        Args:
+            active_tracers (list): A list of :class:`ActiveTracer` objects that
+                encode the metadata for the active tracers.
 
         Returns:
-            :class:`LabelledForm`: a labelled form containing the mass terms.
+            :class:`LabelledForm`: a labelled form containing the mass
+                terms for the active tracers. This is the usual mass form
+                unless using tracer_conservative, where it is multiplied
+                by the reference density.
         """
 
         for i, tracer in enumerate(active_tracers):

From a0f92a0638f03e3300208fd3dc8e654cec78e4f0 Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Fri, 15 Mar 2024 16:34:21 +0000
Subject: [PATCH 16/21] slacken suppress_vertical_wind test and update
 linear_sw_wave KGO

---
 .../data/linear_sw_wave_chkpt.h5              | Bin 573668 -> 580789 bytes
 .../equations/test_coupled_transport.py       |   6 +++---
 .../physics/test_suppress_vertical_wind.py    |   2 +-
 3 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/integration-tests/data/linear_sw_wave_chkpt.h5 b/integration-tests/data/linear_sw_wave_chkpt.h5
index 11420f7c782216a786e9b4b397b8375b9ba3e1f9..03400f12cbae62e62459d63ee14f95ccf31a348a 100644
GIT binary patch
delta 125368
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diff --git a/integration-tests/equations/test_coupled_transport.py b/integration-tests/equations/test_coupled_transport.py
index b314a3236..b64498d32 100644
--- a/integration-tests/equations/test_coupled_transport.py
+++ b/integration-tests/equations/test_coupled_transport.py
@@ -94,10 +94,10 @@ def test_conservative_coupled_transport(tmpdir, m_X_space, tracer_setup):
                                             project_low_method='recover'),
                       'f2': EmbeddedDGOptions()}
         opts = MixedFSOptions(suboptions=suboptions)
-
+    
         transport_scheme = SSPRK3(domain, options=opts, increment_form=False)
-
-    transport_scheme = SSPRK3(domain, increment_form=False)
+    else:
+        transport_scheme = SSPRK3(domain, increment_form=False)
 
     transport_method = [DGUpwind(eqn, 'f1'), DGUpwind(eqn, 'f2')]
 
diff --git a/integration-tests/physics/test_suppress_vertical_wind.py b/integration-tests/physics/test_suppress_vertical_wind.py
index 44f1e05ae..cab6efc76 100644
--- a/integration-tests/physics/test_suppress_vertical_wind.py
+++ b/integration-tests/physics/test_suppress_vertical_wind.py
@@ -92,5 +92,5 @@ def test_suppress_vertical_wind(tmpdir):
     vertical_wind = Function(domain.spaces('theta'))
     vertical_wind.interpolate(dot(u, domain.k))
 
-    tol = 1e-12
+    tol = 1e-10
     assert norm(vertical_wind) < tol

From 498034a407eaf49458460103bd734fa66043974e Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Fri, 15 Mar 2024 16:36:24 +0000
Subject: [PATCH 17/21] lint

---
 integration-tests/equations/test_coupled_transport.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/integration-tests/equations/test_coupled_transport.py b/integration-tests/equations/test_coupled_transport.py
index b64498d32..da474bb4f 100644
--- a/integration-tests/equations/test_coupled_transport.py
+++ b/integration-tests/equations/test_coupled_transport.py
@@ -94,7 +94,7 @@ def test_conservative_coupled_transport(tmpdir, m_X_space, tracer_setup):
                                             project_low_method='recover'),
                       'f2': EmbeddedDGOptions()}
         opts = MixedFSOptions(suboptions=suboptions)
-    
+
         transport_scheme = SSPRK3(domain, options=opts, increment_form=False)
     else:
         transport_scheme = SSPRK3(domain, increment_form=False)

From 4ffee21e1d21c5da8447a0008b7ae4820a4f2046 Mon Sep 17 00:00:00 2001
From: Timothy Andrews <75810156+ta440@users.noreply.github.com>
Date: Mon, 18 Mar 2024 10:07:39 +0000
Subject: [PATCH 18/21] Update gusto/equations.py to add back in non-periodic
 check in no normal flow BCs

Co-authored-by: Thomas Bendall <thomas.bendall@metoffice.gov.uk>
---
 gusto/equations.py | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/gusto/equations.py b/gusto/equations.py
index e0c67f140..b98181d30 100644
--- a/gusto/equations.py
+++ b/gusto/equations.py
@@ -376,7 +376,8 @@ def set_no_normal_flow_bcs(self, domain, no_normal_flow_bc_ids):
                 + 'when there is a variable called "u" and none was found')
 
         Vu = domain.spaces("HDiv")
-        if Vu.extruded:
+        # we only apply no normal-flow BCs when extruded mesh is non periodic
+        if Vu.extruded and not Vu.ufl_domain().topology.extruded_periodic:
             self.bcs['u'].append(DirichletBC(Vu, 0.0, "bottom"))
             self.bcs['u'].append(DirichletBC(Vu, 0.0, "top"))
         for id in no_normal_flow_bc_ids:

From ed99b80563a982ba03821d3a02dac4c65c420cca Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Mon, 18 Mar 2024 12:25:20 +0000
Subject: [PATCH 19/21] better extraction of reference density for
 tracer_conservative transport

---
 gusto/transport_methods.py | 7 +++----
 1 file changed, 3 insertions(+), 4 deletions(-)

diff --git a/gusto/transport_methods.py b/gusto/transport_methods.py
index cd43b921f..93be2c158 100644
--- a/gusto/transport_methods.py
+++ b/gusto/transport_methods.py
@@ -37,13 +37,12 @@ def __init__(self, equation, variable):
         self.transport_equation_type = self.original_form.terms[0].get(transport)
 
         if self.transport_equation_type == TransportEquationType.tracer_conservative:
-            # Extract associated density variable:
-            # This does assume the same ordering of tracers and fields...
-            tracer_idx = self.equation.field_names.index(variable)
-            tracer = self.equation.active_tracers[tracer_idx]
+            # Extract associated density of the variable
+            tracer = next(x for x in self.equation.active_tracers if x.name == variable)
             density_idx = self.equation.field_names.index(tracer.density_name)
             self.conservative_density = split(self.equation.X)[density_idx]
 
+
     def replace_form(self, equation):
         """
         Replaces the form for the transport term in the equation with the

From 0f89ae8733b7dec813955205c2e05f0fa97da07d Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Mon, 18 Mar 2024 12:27:14 +0000
Subject: [PATCH 20/21] lint

---
 gusto/transport_methods.py | 1 -
 1 file changed, 1 deletion(-)

diff --git a/gusto/transport_methods.py b/gusto/transport_methods.py
index 93be2c158..2329015ef 100644
--- a/gusto/transport_methods.py
+++ b/gusto/transport_methods.py
@@ -42,7 +42,6 @@ def __init__(self, equation, variable):
             density_idx = self.equation.field_names.index(tracer.density_name)
             self.conservative_density = split(self.equation.X)[density_idx]
 
-
     def replace_form(self, equation):
         """
         Replaces the form for the transport term in the equation with the

From 52f444df0da8bef1985fe67e7697befc723d0684 Mon Sep 17 00:00:00 2001
From: Tim Andrews <ta440@github.com>
Date: Tue, 19 Mar 2024 16:06:10 +0000
Subject: [PATCH 21/21] made changes suggested by Alex

---
 gusto/time_discretisation.py | 7 +++----
 1 file changed, 3 insertions(+), 4 deletions(-)

diff --git a/gusto/time_discretisation.py b/gusto/time_discretisation.py
index c641a4b5f..a1221c6b9 100644
--- a/gusto/time_discretisation.py
+++ b/gusto/time_discretisation.py
@@ -933,9 +933,8 @@ def rhs(self):
 
             for stage in range(self.nStages):
                 r = self.residual.label_map(
-                    lambda t: t.has_label(time_derivative),
-                    map_if_true=replace_subject(self.field_i[0], old_idx=self.idx),
-                    map_if_false=replace_subject(self.field_i[0], old_idx=self.idx))
+                    all_terms,
+                    map_if_true=replace_subject(self.field_i[0], old_idx=self.idx))
 
                 r = r.label_map(
                     lambda t: t.has_label(time_derivative),
@@ -1000,7 +999,7 @@ def solve_stage(self, x0, stage):
             if self.limiter is not None:
                 self.limiter.apply(self.field_i[stage])
 
-            # Obtain field_ip1 = field_n - dt* sum_m{c_im*F[field_m]}
+            # Obtain field_ip1 = field_n - dt* sum_m{a_im*F[field_m]}
             self.solver[stage].solve()
 
             if (stage == self.nStages - 1):