Docs: Switch integrator to fix wave number example (#2616)

* Docs: Switch integrator to fix wave number example

* tv

Author: DanielDoehring

docs/literate/src/files/differentiable_programming.jl

@@ -230,7 +230,7 @@ function energy_at_final_time(k) # k is the wave number of the initial condition
     semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver,
                                         uEltype = typeof(k))
     ode = semidiscretize(semi, (0.0, 1.0))
-    sol = solve(ode, BS3(); ode_default_options()...)
+    sol = solve(ode, FRK65(), dt = 0.05, adaptive = false, save_everystep = false)
     Trixi.integrate(energy_total, sol.u[end], semi)
 end
 
@@ -280,7 +280,7 @@ function energy_at_final_time(k) # k is the wave number of the initial condition
     semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver,
                                         uEltype = typeof(k))
     ode = semidiscretize(semi, (0.0, 1.0))
-    sol = solve(ode, BS3(); ode_default_options()...)
+    sol = solve(ode, FRK65(), dt = 0.05, adaptive = false, save_everystep = false)
     Trixi.integrate(energy_total, sol.u[end], semi)
 end
 
@@ -330,9 +330,9 @@ semi = SemidiscretizationHyperbolic(mesh, equations, initial_condition, solver,
 # does. This is basically the only part where you need to modify your standard Trixi.jl
 # code to enable automatic differentiation. From there on, the remaining steps
 ode = semidiscretize(semi, (0.0, 1.0))
-sol = solve(ode, BS3(); ode_default_options()...)
+sol = solve(ode, FRK65(), dt = 0.05, adaptive = false, save_everystep = false)
 round(Trixi.integrate(energy_total, sol.u[end], semi), sigdigits = 5)
-@test round(Trixi.integrate(energy_total, sol.u[end], semi), sigdigits = 5) == 0.24986 #src
+@test round(Trixi.integrate(energy_total, sol.u[end], semi), sigdigits = 5) == 0.25 #src
 
 # do not need any modifications since they are sufficiently generic (and enough effort
 # has been spend to allow general types inside these calls).