Refactor PERK2 time integration method to use fixed time step (#1958)

* add fixed time step and test

* fmt

* fix test values

* Update test_tree_1d_advection.jl

* adjustment so that fixed timestep work with savesolcallback when dt is specified

* attempt to avoid argument error

* test with save solution

* add save_solution to elixir

* update test value

---------

Co-authored-by: Daniel Doehring <[email protected]>

examples/tree_1d_dgsem/elixir_advection_perk2.jl

@@ -44,9 +44,15 @@ stepsize_callback = StepsizeCallback(cfl = 2.5)
 
 alive_callback = AliveCallback(alive_interval = analysis_interval)
 
+save_solution = SaveSolutionCallback(dt = 0.1,
+                                     save_initial_solution = true,
+                                     save_final_solution = true,
+                                     solution_variables = cons2prim)
+
 # Create a CallbackSet to collect all callbacks such that they can be passed to the ODE solver
 callbacks = CallbackSet(summary_callback,
                         alive_callback,
+                        save_solution,
                         analysis_callback,
                         stepsize_callback)
 
@@ -59,7 +65,7 @@ callbacks = CallbackSet(summary_callback,
 ode_algorithm = Trixi.PairedExplicitRK2(6, tspan, semi)
 
 sol = Trixi.solve(ode, ode_algorithm,
-                  dt = 1.0, # solve needs some value here but it will be overwritten by the stepsize_callback
+                  dt = 1.0, # Manual time step value, will be overwritten by the stepsize_callback when it is specified.
                   save_everystep = false, callback = callbacks);
 
 # Print the timer summary

src/time_integration/paired_explicit_runge_kutta/methods_PERK2.jl

@@ -180,17 +180,25 @@ function PairedExplicitRK2(num_stages, tspan, eig_vals::Vector{ComplexF64};
 end
 
 # This struct is needed to fake https://github.com/SciML/OrdinaryDiffEq.jl/blob/0c2048a502101647ac35faabd80da8a5645beac7/src/integrators/type.jl#L1
-mutable struct PairedExplicitRKOptions{Callback}
+mutable struct PairedExplicitRKOptions{Callback, TStops}
     callback::Callback # callbacks; used in Trixi
-    adaptive::Bool # whether the algorithm is adaptive; ignored
+    adaptive::Bool # whether the algorithm is adaptive
     dtmax::Float64 # ignored
     maxiters::Int # maximal number of time steps
-    tstops::Vector{Float64} # tstops from https://diffeq.sciml.ai/v6.8/basics/common_solver_opts/#Output-Control-1; ignored
+    tstops::TStops # tstops from https://diffeq.sciml.ai/v6.8/basics/common_solver_opts/#Output-Control-1; ignored
 end
 
 function PairedExplicitRKOptions(callback, tspan; maxiters = typemax(Int), kwargs...)
-    PairedExplicitRKOptions{typeof(callback)}(callback, false, Inf, maxiters,
-                                              [last(tspan)])
+    tstops_internal = BinaryHeap{eltype(tspan)}(FasterForward())
+    # We add last(tspan) to make sure that the time integration stops at the end time
+    push!(tstops_internal, last(tspan))
+    # We add 2 * last(tspan) because add_tstop!(integrator, t) is only called by DiffEqCallbacks.jl if tstops contains a time that is larger than t
+    # (https://github.com/SciML/DiffEqCallbacks.jl/blob/025dfe99029bd0f30a2e027582744528eb92cd24/src/iterative_and_periodic.jl#L92)
+    push!(tstops_internal, 2 * last(tspan))
+    PairedExplicitRKOptions{typeof(callback), typeof(tstops_internal)}(callback,
+                                                                       false, Inf,
+                                                                       maxiters,
+                                                                       tstops_internal)
 end
 
 abstract type PairedExplicitRK end
@@ -207,20 +215,42 @@ mutable struct PairedExplicitRK2Integrator{RealT <: Real, uType, Params, Sol, F,
     du::uType
     u_tmp::uType
     t::RealT
+    tdir::RealT
     dt::RealT # current time step
-    dtcache::RealT # ignored
+    dtcache::RealT # manually set time step
     iter::Int # current number of time steps (iteration)
     p::Params # will be the semidiscretization from Trixi
     sol::Sol # faked
     f::F
     alg::Alg # This is our own class written above; Abbreviation for ALGorithm
     opts::PairedExplicitRKOptions
     finalstep::Bool # added for convenience
+    dtchangeable::Bool
+    force_stepfail::Bool
     # PairedExplicitRK2 stages:
     k1::uType
     k_higher::uType
 end
 
+"""
+    add_tstop!(integrator::PairedExplicitRK2Integrator, t)
+Add a time stop during the time integration process.
+This function is called after the periodic SaveSolutionCallback to specify the next stop to save the solution.
+"""
+function add_tstop!(integrator::PairedExplicitRK2Integrator, t)
+    integrator.tdir * (t - integrator.t) < zero(integrator.t) &&
+        error("Tried to add a tstop that is behind the current time. This is strictly forbidden")
+    # We need to remove the first entry of tstops when a new entry is added.
+    # Otherwise, the simulation gets stuck at the previous tstop and dt is adjusted to zero.
+    if length(integrator.opts.tstops) > 1
+        pop!(integrator.opts.tstops)
+    end
+    push!(integrator.opts.tstops, integrator.tdir * t)
+end
+
+has_tstop(integrator::PairedExplicitRK2Integrator) = !isempty(integrator.opts.tstops)
+first_tstop(integrator::PairedExplicitRK2Integrator) = first(integrator.opts.tstops)
+
 # Forward integrator.stats.naccept to integrator.iter (see GitHub PR#771)
 function Base.getproperty(integrator::PairedExplicitRK, field::Symbol)
     if field === :stats
@@ -241,15 +271,16 @@ function init(ode::ODEProblem, alg::PairedExplicitRK2;
     k_higher = zero(u0)
 
     t0 = first(ode.tspan)
+    tdir = sign(ode.tspan[end] - ode.tspan[1])
     iter = 0
 
-    integrator = PairedExplicitRK2Integrator(u0, du, u_tmp, t0, dt, zero(dt), iter,
+    integrator = PairedExplicitRK2Integrator(u0, du, u_tmp, t0, tdir, dt, dt, iter,
                                              ode.p,
                                              (prob = ode,), ode.f, alg,
                                              PairedExplicitRKOptions(callback,
                                                                      ode.tspan;
                                                                      kwargs...),
-                                             false,
+                                             false, true, false,
                                              k1, k_higher)
 
     # initialize callbacks
@@ -301,6 +332,8 @@ function step!(integrator::PairedExplicitRK2Integrator)
         error("time step size `dt` is NaN")
     end
 
+    modify_dt_for_tstops!(integrator)
+
     # if the next iteration would push the simulation beyond the end time, set dt accordingly
     if integrator.t + integrator.dt > t_end ||
        isapprox(integrator.t + integrator.dt, t_end)
@@ -383,17 +416,40 @@ u_modified!(integrator::PairedExplicitRK, ::Bool) = false
 
 # used by adaptive timestepping algorithms in DiffEq
 function set_proposed_dt!(integrator::PairedExplicitRK, dt)
-    integrator.dt = dt
+    (integrator.dt = dt; integrator.dtcache = dt)
 end
 
 function get_proposed_dt(integrator::PairedExplicitRK)
-    return integrator.dt
+    return ifelse(integrator.opts.adaptive, integrator.dt, integrator.dtcache)
 end
 
 # stop the time integration
 function terminate!(integrator::PairedExplicitRK)
     integrator.finalstep = true
-    empty!(integrator.opts.tstops)
+end
+
+"""
+    modify_dt_for_tstops!(integrator::PairedExplicitRK)
+Modify the time-step size to match the time stops specified in integrator.opts.tstops.
+To avoid adding OrdinaryDiffEq to Trixi's dependencies, this routine is a copy of
+https://github.com/SciML/OrdinaryDiffEq.jl/blob/d76335281c540ee5a6d1bd8bb634713e004f62ee/src/integrators/integrator_utils.jl#L38-L54
+"""
+function modify_dt_for_tstops!(integrator::PairedExplicitRK)
+    if has_tstop(integrator)
+        tdir_t = integrator.tdir * integrator.t
+        tdir_tstop = first_tstop(integrator)
+        if integrator.opts.adaptive
+            integrator.dt = integrator.tdir *
+                            min(abs(integrator.dt), abs(tdir_tstop - tdir_t)) # step! to the end
+        elseif iszero(integrator.dtcache) && integrator.dtchangeable
+            integrator.dt = integrator.tdir * abs(tdir_tstop - tdir_t)
+        elseif integrator.dtchangeable && !integrator.force_stepfail
+            # always try to step! with dtcache, but lower if a tstop
+            # however, if force_stepfail then don't set to dtcache, and no tstop worry
+            integrator.dt = integrator.tdir *
+                            min(abs(integrator.dtcache), abs(tdir_tstop - tdir_t)) # step! to the end
+        end
+    end
 end
 
 # used for AMR (Adaptive Mesh Refinement)

test/test_tree_1d_advection.jl

@@ -84,8 +84,28 @@ end
 
 @trixi_testset "elixir_advection_perk2.jl" begin
     @test_trixi_include(joinpath(EXAMPLES_DIR, "elixir_advection_perk2.jl"),
-                        l2=[0.014139242834192841],
-                        linf=[0.01999756655819429])
+                        l2=[0.011288030389423475],
+                        linf=[0.01596735472556976])
+    # Ensure that we do not have excessive memory allocations
+    # (e.g., from type instabilities)
+    let
+        t = sol.t[end]
+        u_ode = sol.u[end]
+        du_ode = similar(u_ode)
+        @test (@allocated Trixi.rhs!(du_ode, u_ode, semi, t)) < 8000
+    end
+end
+
+# Testing the second-order paired explicit Runge-Kutta (PERK) method without stepsize callback
+@trixi_testset "elixir_advection_perk2.jl(fixed time step)" begin
+    @test_trixi_include(joinpath(EXAMPLES_DIR, "elixir_advection_perk2.jl"),
+                        dt=2.0e-3,
+                        tspan=(0.0, 20.0),
+                        save_solution=SaveSolutionCallback(dt = 0.1 + 1.0e-8),
+                        callbacks=CallbackSet(summary_callback, save_solution,
+                                              analysis_callback, alive_callback),
+                        l2=[9.886271430207691e-6],
+                        linf=[3.729460413781638e-5])
     # Ensure that we do not have excessive memory allocations
     # (e.g., from type instabilities)
     let