Add MREEF multirate extrapolated Euler method with tests and proper integration

Author: Harsh Singh

lib/OrdinaryDiffEqLowOrderRK/src/OrdinaryDiffEqLowOrderRK.jl

@@ -1,45 +1,22 @@
 module OrdinaryDiffEqLowOrderRK
 
-import OrdinaryDiffEqCore:
-    alg_order,
-    isfsal,
-    beta2_default,
-    beta1_default,
+import OrdinaryDiffEqCore: alg_order, isfsal, beta2_default, beta1_default,
     alg_stability_size,
-    ssp_coefficient,
-    OrdinaryDiffEqAlgorithm,
+    ssp_coefficient, OrdinaryDiffEqAlgorithm,
     OrdinaryDiffEqExponentialAlgorithm,
-    explicit_rk_docstring,
-    generic_solver_docstring,
+    explicit_rk_docstring, generic_solver_docstring,
     trivial_limiter!,
     OrdinaryDiffEqAdaptiveAlgorithm,
-    unwrap_alg,
-    initialize!,
-    perform_step!,
+    unwrap_alg, initialize!, perform_step!,
     calculate_residuals,
-    calculate_residuals!,
-    _ode_addsteps!,
-    @OnDemandTableauExtract,
+    calculate_residuals!, _ode_addsteps!, @OnDemandTableauExtract,
     constvalue,
-    OrdinaryDiffEqMutableCache,
-    uses_uprev,
-    OrdinaryDiffEqConstantCache,
-    @fold,
-    @cache,
-    CompiledFloats,
-    alg_cache,
-    CompositeAlgorithm,
-    AutoAlgSwitch,
-    _ode_interpolant,
-    _ode_interpolant!,
-    full_cache,
-    accept_step_controller,
-    DerivativeOrderNotPossibleError,
-    du_cache,
-    u_cache,
-    get_fsalfirstlast,
-    copyat_or_push!,
-    _unwrap_val
+    OrdinaryDiffEqMutableCache, uses_uprev,
+    OrdinaryDiffEqConstantCache, @fold,
+    @cache, CompiledFloats, alg_cache, CompositeAlgorithm,
+    AutoAlgSwitch, _ode_interpolant, _ode_interpolant!, full_cache,
+    accept_step_controller, DerivativeOrderNotPossibleError,
+    du_cache, u_cache, get_fsalfirstlast, copyat_or_push!, _unwrap_val
 using SciMLBase
 import MuladdMacro: @muladd
 import FastBroadcast: @..
@@ -64,33 +41,10 @@ include("low_order_rk_addsteps.jl")
 include("split_perform_step.jl")
 include("fixed_timestep_perform_step.jl")
 
-export Euler,
-    SplitEuler,
-    MREEF,
-    Heun,
-    Ralston,
-    Midpoint,
-    RK4,
-    BS3,
-    OwrenZen3,
-    OwrenZen4,
-    OwrenZen5,
-    BS5,
-    DP5,
-    Anas5,
-    RKO65,
-    FRK65,
-    RKM,
-    MSRK5,
-    MSRK6,
-    PSRK4p7q6,
-    PSRK3p5q4,
-    PSRK3p6q5,
-    Stepanov5,
-    SIR54,
-    Alshina2,
-    Alshina3,
-    Alshina6,
-    AutoDP5
+export Euler, SplitEuler, MREEF, Heun, Ralston, Midpoint, RK4,
+    BS3, OwrenZen3, OwrenZen4, OwrenZen5, BS5,
+    DP5, Anas5, RKO65, FRK65, RKM, MSRK5, MSRK6,
+    PSRK4p7q6, PSRK3p5q4, PSRK3p6q5, Stepanov5, SIR54,
+    Alshina2, Alshina3, Alshina6, AutoDP5
 
 end

lib/OrdinaryDiffEqLowOrderRK/src/alg_utils.jl

@@ -41,7 +41,11 @@ alg_stability_size(alg::DP5) = 3.3066
 
 ssp_coefficient(alg::Euler) = 1
 
-function prepare_alg(alg::SplitEuler, u0::AbstractArray, p, prob)
+function prepare_alg(
+        alg::SplitEuler,
+        u0::AbstractArray,
+        p, prob
+    )
     return alg
 end