diff --git a/src/time_integration/paired_explicit_runge_kutta/methods_PERK2.jl b/src/time_integration/paired_explicit_runge_kutta/methods_PERK2.jl
index ad385e6df2..bad40c6156 100644
--- a/src/time_integration/paired_explicit_runge_kutta/methods_PERK2.jl
+++ b/src/time_integration/paired_explicit_runge_kutta/methods_PERK2.jl
@@ -106,7 +106,7 @@ function compute_PairedExplicitRK2_butcher_tableau(num_stages,
 end
 
 @doc raw"""
-    PairedExplicitRK2(num_stages, base_path_monomial_coeffs::AbstractString, dt_opt,
+    PairedExplicitRK2(num_stages, base_path_monomial_coeffs::AbstractString, dt_opt = nothing,
                       bS = 1.0, cS = 0.5)
     PairedExplicitRK2(num_stages, tspan, semi::AbstractSemidiscretization;
                       verbose = false, bS = 1.0, cS = 0.5)
@@ -117,7 +117,8 @@ end
     - `base_path_monomial_coeffs` (`AbstractString`): Path to a file containing 
       monomial coefficients of the stability polynomial of PERK method.
       The coefficients should be stored in a text file at `joinpath(base_path_monomial_coeffs, "gamma_$(num_stages).txt")` and separated by line breaks.
-    - `dt_opt` (`Float64`): Optimal time step size for the simulation setup.
+    - `dt_opt` (`Float64`, optional): Optimal time step size for the simulation setup. Can be `nothing` if it is unknown. 
+       In this case the optimal CFL number cannot be computed and the [`StepsizeCallback`](@ref) cannot be used.
     - `tspan`: Time span of the simulation.
     - `semi` (`AbstractSemidiscretization`): Semidiscretization setup.
     -  `eig_vals` (`Vector{ComplexF64}`): Eigenvalues of the Jacobian of the right-hand side (rhs) of the ODEProblem after the
@@ -135,6 +136,8 @@ optimized for a certain simulation setup (PDE, IC & BC, Riemann Solver, DG Solve
 - Brian Vermeire (2019).
   Paired explicit Runge-Kutta schemes for stiff systems of equations
   [DOI: 10.1016/j.jcp.2019.05.014](https://doi.org/10.1016/j.jcp.2019.05.014)
+
+Note: To use this integrator, the user must import the `Convex` and `ECOS` packages.
 """
 mutable struct PairedExplicitRK2 <: AbstractPairedExplicitRKSingle
     const num_stages::Int
@@ -144,12 +147,12 @@ mutable struct PairedExplicitRK2 <: AbstractPairedExplicitRKSingle
     b1::Float64
     bS::Float64
     cS::Float64
-    dt_opt::Float64
+    dt_opt::Union{Float64, Nothing}
 end # struct PairedExplicitRK2
 
 # Constructor that reads the coefficients from a file
 function PairedExplicitRK2(num_stages, base_path_monomial_coeffs::AbstractString,
-                           dt_opt,
+                           dt_opt = nothing,
                            bS = 1.0, cS = 0.5)
     # If the user has the monomial coefficients, they also must have the optimal time step
     a_matrix, c = compute_PairedExplicitRK2_butcher_tableau(num_stages,
@@ -246,6 +249,10 @@ function calculate_cfl(ode_algorithm::AbstractPairedExplicitRKSingle, ode)
     semi = ode.p
     dt_opt = ode_algorithm.dt_opt
 
+    if isnothing(dt_opt)
+        error("The optimal time step `dt_opt` must be provided.")
+    end
+
     mesh, equations, solver, cache = mesh_equations_solver_cache(semi)
     u = wrap_array(u_ode, mesh, equations, solver, cache)
 
diff --git a/test/test_unit.jl b/test/test_unit.jl
index 4e8945eab8..523ce7061e 100644
--- a/test/test_unit.jl
+++ b/test/test_unit.jl
@@ -1681,7 +1681,7 @@ end
     Trixi.download("https://gist.githubusercontent.com/DanielDoehring/8db0808b6f80e59420c8632c0d8e2901/raw/39aacf3c737cd642636dd78592dbdfe4cb9499af/MonCoeffsS6p2.txt",
                    joinpath(path_coeff_file, "gamma_6.txt"))
 
-    ode_algorithm = Trixi.PairedExplicitRK2(6, path_coeff_file, 42) # dummy optimal time step (dt_opt plays no role in determining `a_matrix`)
+    ode_algorithm = Trixi.PairedExplicitRK2(6, path_coeff_file)
 
     @test isapprox(ode_algorithm.a_matrix,
                    [0.12405417889682908 0.07594582110317093