Try using fancier initial guess for implicit solver

Author: dennisYatunin

src/solvers/ark_algorithm.jl

@@ -274,6 +274,7 @@ function init_cache(prob::DiffEqBase.AbstractODEProblem, alg::ARKAlgorithm; kwar
         ΔU_imp_nz = dense_tuple(ΔU_imp_nz_sparse, s + 1, nothing),
         u_on_stage = similar(u0),
         u_plus_Δu_lim = similar(u0),
+        T_imp_estimate = similar(u0),
     )
 
     newtons_method_cache = if !iszero(Γ) && !isnothing(T_imp!)
@@ -315,6 +316,7 @@ function step_u!(integrator, cache::ARKAlgorithmCache)
         ΔU_imp_nz,
         u_on_stage,
         u_plus_Δu_lim,
+        T_imp_estimate,
     ) = cache.timestepper_cache
 
     Δt = integrator.dt
@@ -382,8 +384,11 @@ function step_u!(integrator, cache::ARKAlgorithmCache)
         # Compute the state on the current stage. Apply post_stage! if it is
         # different from the previous state.
         if !isnothing(T_imp!) && !iszero(Δtγ)
+            @. u_on_stage = u_minus_Δu_imp_from_solve
+            # T_imp!(T_imp_estimate, u_on_stage, p, t_imp)
+            # @. u_on_stage += Δtγ * T_imp_estimate
+
             # Solve u′ ≈ u_minus_Δu_imp_from_solve + Δtγ * T_imp(u′, p, t_imp).
-            @. u_on_stage = u # TODO: Is this a good idea?
             solve_newton!(
                 newtons_method,
                 newtons_method_cache,