Add constructor of PERK3

src/time_integration/paired_explicit_runge_kutta/methods_PERK2.jl

@@ -2,7 +2,6 @@
 # Since these FMAs can increase the performance of many numerical algorithms,
 # we need to opt-in explicitly.
 # See https://ranocha.de/blog/Optimizing_EC_Trixi for further details.
-include("polynomial_optimizer.jl")
 @muladd begin
 
 abstract type PERK end
@@ -166,7 +165,7 @@
   callback::Callback # callbacks; used in Trixi
   adaptive::Bool # whether the algorithm is adaptive; ignored
   dtmax::Float64 # ignored
   maxiters::Int # maximal numer of time steps
   tstops::Vector{Float64} # tstops from https://diffeq.sciml.ai/v6.8/basics/common_solver_opts/#Output-Control-1; ignored
 end
 

src/time_integration/paired_explicit_runge_kutta/methods_PERK3.jl

@@ -2,43 +2,152 @@
 # Since these FMAs can increase the performance of many numerical algorithms,
 # we need to opt-in explicitly.
 # See https://ranocha.de/blog/Optimizing_EC_Trixi for further details.
+using NLsolve
 @muladd begin
 
-function ComputePERK3_ButcherTableau(NumStages::Int, BasePathMonCoeffs::AbstractString, cS2::Float64)
+function f_own(a_unknown, num_stages, num_stage_evals, mon_coeffs)
+    c_s2 = 1 # c_s2 = c_ts(S-2)
+    c_ts = c_own(c_s2, num_stages) # ts = timestep
+
+    c_eq = zeros(num_stage_evals - 2) # Add equality constraint that c_s2 is equal to 1
+    # Both terms should be present
+    for i in 1:(num_stage_evals - 4)
+        term1 = a_unknown[num_stage_evals - 1]
+        term2 = a_unknown[num_stage_evals]
+        for j in 1:i
+            term1 *= a_unknown[num_stage_evals - 1 - j]
+            term2 *= a_unknown[num_stage_evals - j]
+        end
+        term1 *= c_ts[num_stages - 2 - i] * 1/6
+        term2 *= c_ts[num_stages - 1 - i] * 4/6
 
-  # c Vector form Butcher Tableau (defines timestep per stage)
-  c = zeros(NumStages)
-  for k in 2:NumStages-2
-    c[k] = cS2 * (k - 1)/(NumStages - 3) # Equidistant timestep distribution (similar to PERK2)
-  end
+        c_eq[i] = mon_coeffs[i] - (term1 + term2)
+    end
+
+    # Highest coefficient: Only one term present
+    i = num_stage_evals - 3
+    term2 = a_unknown[num_stage_evals]
+    for j in 1:i
+        term2 *= a_unknown[num_stage_evals - j]
+    end
+    term2 *= c_ts[num_stages - 1 - i] * 4/6
+
+    c_eq[i] = mon_coeffs[i] - term2
+
+    c_eq[num_stage_evals - 2] = 1.0 - 4 * a_unknown[num_stage_evals] - a_unknown[num_stage_evals - 1]
+
+    return c_eq
+end
+
+function c_own(c_s2, num_stages)
+    c_ts = zeros(num_stages)
+
+    # Last timesteps as for SSPRK33
+    c_ts[num_stages]     = 0.5
+    c_ts[num_stages - 1] = 1
+
+    # Linear increasing timestep for remainder
+    for i in 2:num_stages-2
+        c_ts[i] = c_s2 * (i-1)/(num_stages - 3)
+    end
+    return c_ts
+end
+
+function compute_PERK3_butcher_tableau(num_stages, num_stage_evals, semi::AbstractSemidiscretization)
+
+    # Initialize array of c
+    c_ts = c_own(1.0, num_stages)
+
+    # Initialize the array of our solution
+    a_unknown = zeros(num_stage_evals)
+
+    # Special case of e = 3
+    if num_stage_evals == 3
+        a_unknown = [0, c_ts[2], 0.25]
+
+    else
+        # Calculate coefficients of the stability polynomial in monomial form
+        cons_order = 3
+        dt_max = 1.0
+        dt_eps = 1e-9
+        filter_threshold = 1e-12
+
+        # Compute spectrum
+        J = jacobian_ad_forward(semi)
+        eig_vals = eigvals(J)
+        num_eig_vals, eig_vals = filter_eigvals(eig_vals, filter_threshold)
+
+        mon_coeffs, dt, _ = bisection(cons_order, num_eig_vals, num_stages, dt_max, dt_eps, eig_vals)
+        mon_coeffs = undo_normalization!(cons_order, num_stages, mon_coeffs)
+
+        # Define the objective_function
+        function objective_function(x)
+            return f_own(x, num_stages, num_stage_evals, mon_coeffs)
+        end
+
+        # Call nlsolver to solve repeatedly until the result is not NaN or negative values
+        is_sol_valid = false
+        while !is_sol_valid
+            # Initialize initial guess
+            x0 = 0.1 .* rand(num_stage_evals)
+            x0[1] = 0.0
+            x0[2] = c_ts[2]
+
+            sol = nlsolve(objective_function, x0, method = :trust_region, ftol = 1e-15, iterations = 10^4, xtol = 1e-13)
+
+            a_unknown = sol.zero
+
+            # Check if the values a[i, i-1] >= 0.0 (which stem from the nonlinear solver) and subsequently c[i] - a[i, i-1] >= 0.0
+            is_sol_valid = all(x -> !isnan(x) && x >= 0, a_unknown[3:end]) && all(x -> !isnan(x) && x >= 0 , c_ts[3:end] .- a_unknown[3:end])
+        end
+    end
+
+    println("a_unknown")
+    println(a_unknown[3:end]) # To debug
+
+    a_matrix = zeros(num_stages - 2, 2)
+    a_matrix[:, 1] = c_ts[3:end]
+    a_matrix[:, 1] -= a_unknown[3:end]
+    a_matrix[:, 2]  = a_unknown[3:end]
+
+    return a_matrix, c_ts
+end
+
+function compute_PERK3_butcher_tableau(num_stages::Int, base_path_mon_coeffs::AbstractString, c_s2::Float64)
+
+    # c Vector form Butcher Tableau (defines timestep per stage)
+    c = zeros(num_stages)
+    for k in 2:num_stages-2
+      c[k] = c_s2 * (k - 1)/(num_stages - 3) # Equidistant timestep distribution (similar to PERK2)
+    end
 
-  # Original third order proposed PERK
-  #=
-  c[NumStages - 1] = 1.0/3.0
-  c[NumStages]     = 1.0
-  =#
-  # Own third order PERK based on SSPRK33
-  c[NumStages - 1] = 1.0
-  c[NumStages]     = 0.5
-
-  println("Timestep-split: "); display(c); println("\n")
+    # Original third order proposed PERK
+    #=
+    c[num_stages - 1] = 1.0/3.0
+    c[num_stages]     = 1.0
+    =#
+    # Own third order PERK based on SSPRK33
+    c[num_stages - 1] = 1.0
+    c[num_stages]     = 0.5
+
+    println("Timestep-split: "); display(c); println("\n")
 
-  # - 2 Since First entry of A is always zero (explicit method) and second is given by c_2 (consistency)
-  CoeffsMax = NumStages - 2
+    # - 2 Since First entry of A is always zero (explicit method) and second is given by c_2 (consistency)
+    coeffs_max = num_stages - 2
 
-  AMatrix = zeros(CoeffsMax, 2)
-  AMatrix[:, 1] = c[3:end]
+    a_matrix = zeros(coeffs_max, 2)
+    a_matrix[:, 1] = c[3:end]
 
-  PathMonCoeffs = BasePathMonCoeffs * "a_" * string(NumStages) * "_" * string(NumStages) * ".txt"
-  NumMonCoeffs, A = read_file(PathMonCoeffs, Float64)
-  @assert NumMonCoeffs == CoeffsMax
+    path_mon_coeffs = base_path_mon_coeffs * "a_" * string(num_stages) * "_" * string(num_stages) * ".txt"
+    num_mon_coeffs, A = read_file(path_mon_coeffs, Float64)
+    @assert num_mon_coeffs == coeffs_max
 
-  AMatrix[:, 1] -= A
-  AMatrix[:, 2]  = A
+    a_matrix[:, 1] -= A
+    a_matrix[:, 2]  = A
 
-  println("A matrix: "); display(AMatrix); println()
+    println("A matrix: "); display(AMatrix); println()
 
-  return AMatrix, c
+    return a_matrix, c
 end
 
 """
@@ -53,20 +162,32 @@ CarpenterKennedy2N{54, 43} methods.
 """
 
 mutable struct PERK3 <: PERKSingle
-  const NumStages::Int
+    const num_stages::Int
+
+    a_matrix::Matrix{Float64}
+    c::Vector{Float64}
 
-  AMatrix::Matrix{Float64}
-  c::Vector{Float64}
+    # Constructor for previously computed A Coeffs
+    function PERK3(num_stages_::Int, base_path_mon_coeffs_::AbstractString, c_s2_::Float64=1.0)
+        newPERK3 = new(num_stages_)
 
-  # Constructor for previously computed A Coeffs
-  function PERK3(NumStages_::Int, BasePathMonCoeffs_::AbstractString, cS2_::Float64=1.0)
+        newPERK3.a_matrix, newPERK3.c = compute_PERK3_butcher_tableau(num_stages_, base_path_mon_coeffs_, c_s2_)
+
+        return newPERK3
+    end
 
-    newPERK3 = new(NumStages_)
+    # Constructor that computes Butcher matrix A coefficients from a semidiscretization
+    function PERK3(num_stages_::Int, num_stage_evals_ ::Int, semi_::AbstractSemidiscretization)
+
+        newPERK3 = new(num_stages_)
+
+        newPERK3.a_matrix, newPERK3.c = compute_PERK3_butcher_tableau(num_stages_, num_stage_evals_, semi_)
+
+        display(newPERK3.a_matrix)
+
+        return newPERK3
+    end
 
-    newPERK3.AMatrix, newPERK3.c = 
-      ComputePERK3_ButcherTableau(NumStages_, BasePathMonCoeffs_, cS2_)
-    return newPERK3
-  end
 end # struct PERK3
 
 

src/time_integration/paired_explicit_runge_kutta/paired_explicit_runge_kutta.jl

@@ -7,5 +7,6 @@
 
 include("methods_PERK2.jl")
 include("methods_PERK3.jl")
+include("polynomial_optimizer.jl")
 
 end # @muladd