Burgers' Equation Finite-Difference Method Work-Precision Diagrams

Problem Description

The Burgers' partial differential equation is solved on the domain $[0, L] \times [0, T] \in \mathbb R \times R,~L = 1,~T = 1$, with the following initial and boundary conditions:

\[\begin{aligned} \partial_t u(t,x) & = -\partial_x u^2(t,x) + \nu \partial_x^2 u(t,x), \\ u(0,x) & = \exp\left[\frac{(x - 0.5)^2}{2 \times 0.05^2}\right], \\ u(-L,x) & = u(L,x) = 0. \end{aligned}\]

The spatial derivative operators are represented via finite difference approximations on an equispaced grid of points $x_s \in [-L, L]$. The diffusion coefficient $\nu = 10^{-2}$ is chosen to produce `interesting' behavior as seen in the reference solution below.

\[\begin{aligned} \frac{du}{dt} & = -D_x u^2(t,x) + \nu D_x^2 u(t,x), \\ u(0,x_s) & = \exp\left[\frac{(x_s - 0.5)^2}{2 \times 0.05^2}\right], \\ u(t,-L) & = u(t,L) = 0. \end{aligned}\]

Implementation

using OrdinaryDiffEq
using OrdinaryDiffEqBDF, OrdinaryDiffEqExponentialRK, OrdinaryDiffEqIMEXMultistep, OrdinaryDiffEqMultirate, OrdinaryDiffEqRosenbrock, OrdinaryDiffEqSDIRK
using DiffEqDevTools
using SciMLOperators
using LinearSolve
using LinearAlgebra
using SparseArrays
using Sundials
using SummationByPartsOperators
const SBP = SummationByPartsOperators
using Plots
gr();

nonlinear_convection!(du, u, p, t) = du .= (-p.alpha / 3) * (u .* (p.D1 * u) + p.D1 * (u .^ 2))

# Constructor
function burgers(N, L)
    # Derivative operators
    D1 = derivative_operator(MattssonSvärdNordström2004(); 
                             derivative_order = 1, accuracy_order = 2, 
                             xmin = -L, xmax = L, N = N)
    D2 = derivative_operator(MattssonSvärdNordström2004(); 
                             derivative_order = 2, accuracy_order = 2, 
                             xmin = -L, xmax = L, N = N)
    
    x = LinRange(0, L, N) # Domain discretization
    u0 = @. exp(-(x - 0.5)^2 / (2 * 0.05^2)) # Initial condition
    nu = 1e-2 # Diffusion coefficient
    alpha = 1.0 # Convection coefficient
    p = (; D1, alpha)

    prob = SplitODEProblem(MatrixOperator(nu * sparse(D2)), nonlinear_convection!, u0, (0.0, 1.0), p)

    return x, prob
end;

Reference Solution

Using an adaptive timestepping method to solve the system of ordinary differential equations with high precision.

N = 128
L = 1.0
xs, prob = burgers(N, L)
sol = solve(prob, AutoVern7(Rodas5(autodiff=AutoFiniteDiff())); abstol=1e-14, reltol=1e-14, adaptive=true)

test_sol = TestSolution(sol); # Reference solution for error estimation

tslices = LinRange(prob.tspan..., 50)
ys = mapreduce(sol, hcat, tslices)
p = heatmap(xs, tslices, ys', xlabel="x", ylabel="t")

Work-Precision Diagrams

High Tolerances

Implicit-Explicit Methods

abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (1:4)
multipliers = 0.5 .^ (0:3)
setups = [
    Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers),
    Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers),
    Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers),
    Dict(:alg => SBDF2(), :dts => 1e-4 * multipliers),
]
labels = hcat(
    "IMEXEuler",
    "CNAB2",
    "CNLF2",
    "SBDF2",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    print_names=true, names=labels, numruns=5, error_estimate=:l2,
    save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));

plot(wp, label=labels, markershape=:auto, title="Work-Precision Diagram, High Tolerance")

IMEXEuler
CNAB2
CNLF2
SBDF2
118.938464 seconds (61.19 M allocations: 20.618 GiB, 6.77% gc time, 26.89% 
compilation time)

Exponential Integrators

abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (1:4)
multipliers = 0.5 .^ (0:3)
setups = [
    Dict(:alg => NorsettEuler(), :dts => 1e-4 * multipliers),
    Dict(:alg => NorsettEuler(krylov=true, m=5), :dts => 1e-4 * multipliers),
    Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-4 * multipliers),
    Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-3 * multipliers),
    Dict(:alg => HochOst4(), :dts => 1e-3 * multipliers),
    Dict(:alg => HochOst4(krylov=true, m=20), :dts => 1e-3 * multipliers),
    Dict(:alg => Friedli(), :dts => 1e-3 * multipliers),
    Dict(:alg => Friedli(krylov=true, m=20), :dts => 1e-3 * multipliers)
]
labels = hcat(
    "NorsettEuler (caching)", 
    "NorsettEuler (m=5)",
    "NorsettEuler (m=20)",
    "ETDRK2 (caching)", 
    "ETDRK2 (m=5)", 
    "ETDRK2 (m=20)",
    "HochOst4 (caching)",
    "HochOst4 (m=20)",
    "Friedli (caching)",
    "Friedli (m=20)"
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    print_names=true, names=labels, numruns=5, error_estimate=:l2,
    save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));

plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, High Tolerance")

NorsettEuler (caching)
NorsettEuler (m=5)
NorsettEuler (m=20)
ETDRK2 (caching)
ETDRK2 (m=5)
ETDRK2 (m=20)
HochOst4 (caching)
HochOst4 (m=20)
Friedli (caching)
Friedli (m=20)
307.067568 seconds (110.30 M allocations: 66.170 GiB, 5.62% gc time, 9.22% 
compilation time)

Multirate Methods

The methods from OrdinaryDiffEqMultirate.jl consume the same SplitODEProblem: the first component (the stiff linear diffusion) is advanced with m explicit substeps inside every macro step, while the second component (the nonlinear convection) is evaluated only at the macro rate. Multirate integration pays off when the slow component is expensive relative to the fast one. That is not the case here, since both terms cost a sparse matrix-vector product, so the substeps buy stability for the diffusion term without saving any slow work, and the comparison against CNAB2 and ETDRK2 from the sections above measures the price of buying stability this way rather than implicitly or exactly.

The linear operator has spectral radius about 161 and the macro steps below go up to 2e-2, so m = 4 puts the largest micro step at 2e-2 * 161 / 4, about 0.8, half of the explicit micro stability bound. In a scan from m = 2 to m = 8 the errors changed by under two percent at every step size, so the macro error dominates and larger m only adds micro work. The multirate methods are adaptive by default but are run at fixed macro steps here, matching how the other fixed step methods in this file are measured. The implicit members of the family (MRIGARKIRK21a, MRIGARKESDIRK34a, MRIGARKESDIRK46a) treat the slow component implicitly and target a stiff slow part; the diffusion here is only mildly stiff, so they are benchmarked on the Kuramoto-Sivashinsky problem instead, where the linear operator is stiff enough for the implicit treatment to matter.

For cross family context the same five reference methods appear in every multirate figure of this folder: adaptive Tsit5, Rodas5P, and FBDF on the joined right hand side (FBDF cannot consume the split form), KenCarp4 on the split problem, and fixed step ETDRK4 at the step sizes this file uses for it elsewhere. CNAB2 and ETDRK2 from the sections above stay in as local references.

abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (1:4)
multipliers = 0.5 .^ (0:3)
# The joined problem calls f1 and f2 through its own buffer: a direct call of the
# SplitFunction uses an internal scratch cache that aliases prob.u0, so it would
# silently overwrite the initial condition of every later solve in the sweep.
joined_rhs! = let stiff_cache = zero(prob.u0)
    (du, u, p, t) -> begin
        prob.f.f1(stiff_cache, u, p, t)
        prob.f.f2(du, u, p, t)
        du .+= stiff_cache
    end
end
prob_joined = ODEProblem(joined_rhs!, copy(prob.u0), prob.tspan, prob.p)
setups = [
    Dict(:alg => MREEF(m = 4, order = 4), :adaptive => false, :dts => 2e-2 * multipliers),
    Dict(:alg => MRAB(k = 2, m = 4), :adaptive => false, :dts => 2e-2 * multipliers),
    Dict(:alg => MIS(m = 4), :adaptive => false, :dts => 2e-2 * multipliers),
    Dict(:alg => MRIGARKERK22a(m = 4), :adaptive => false, :dts => 2e-2 * multipliers),
    Dict(:alg => MRIGARKERK33a(m = 4), :adaptive => false, :dts => 2e-2 * multipliers),
    Dict(:alg => MRIGARKERK45a(m = 4), :adaptive => false, :dts => 2e-2 * multipliers),
    Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers),
    Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers),
    Dict(:alg => KenCarp4()),
    Dict(:alg => Tsit5(), :prob_choice => 2),
    Dict(:alg => Rodas5P(autodiff = AutoFiniteDiff()), :prob_choice => 2),
    Dict(:alg => FBDF(autodiff = AutoFiniteDiff()), :prob_choice => 2),
    Dict(:alg => NordsieckBDF(autodiff = AutoFiniteDiff()), :prob_choice => 2),
]
labels = hcat(
    "MREEF",
    "MRAB",
    "MIS",
    "MRIGARKERK22a",
    "MRIGARKERK33a",
    "MRIGARKERK45a",
    "CNAB2",
    "ETDRK2",
    "ETDRK4",
    "KenCarp4",
    "Tsit5",
    "Rodas5P",
    "FBDF","NordsieckBDF",
)
@time wp = WorkPrecisionSet([prob, prob_joined], abstols, reltols, setups;
    print_names=true, names=labels, numruns=5, error_estimate=:l2,
    save_everystep=false, appxsol=[test_sol, test_sol], maxiters=Int(1e5));

plot(wp, label=labels, markershape=:auto, title="Multirate Methods, High Tolerance")

MREEF
MRAB
MIS
MRIGARKERK22a
MRIGARKERK33a
MRIGARKERK45a
CNAB2
ETDRK2
ETDRK4
KenCarp4
Tsit5
Rodas5P
FBDF
NordsieckBDF
135.922375 seconds (58.03 M allocations: 25.213 GiB, 4.34% gc time, 39.03% 
compilation time)

The cross family view narrows the earlier reading of this figure. Adaptive Tsit5 on the joined problem is the fastest method in the practical middle of the range: it holds a near constant 0.4 to 0.5 milliseconds per solve while its error falls from 9e-3 to 1.3e-7, because the mild diffusion caps its step at the stability limit and tightening the tolerance costs it nothing until error control takes over. FBDF runs a clean curve to 2e-6 at about 3 milliseconds and Rodas5P to 1.2e-7 at about 12 milliseconds. The multirate members' real estate is the tight end: MREEF reaches 2.2e-10 in about 10 milliseconds and MRIGARKERK45a matches it at twice the cost, errors that no adaptive reference here touches, with fixed step ETDRK4 the only method going deeper (1.1e-11 at a quarter of a second, 25 times slower than MREEF at matched 2e-10). That tight end margin is still a statement about step economics rather than multirate structure: the macro steps are 200 times larger than the fixed steps the older references run at, and four cheap substeps keep the diffusion stable, so any stabilized explicit treatment of this split would do as well. KenCarp4 only reaches 1e-4 at these pairs and is slower than every multirate member at matched error. MRAB remains the weak member: freezing the slow rate over the macro step caps it at first order and holds its error above 7e-4 across the sweep.

Comparisons Between Families

abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (1:4)
multipliers = 0.5 .^ (0:3)
setups = [
    Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers),
    Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers),
    Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers),
]
labels = hcat(
    "CNAB2 (dense linsolve)",
    "CNAB2 (Krylov linsolve)", 
    "ETDRK2 (caching)",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    print_names=true, names=labels, numruns=5, error_estimate=:l2,
    save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));

plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances")

CNAB2 (dense linsolve)
CNAB2 (Krylov linsolve)
ETDRK2 (caching)
 80.695674 seconds (31.59 M allocations: 29.724 GiB, 15.75% gc time, 4.25% 
compilation time)

Low Tolerances

Implicit-Explicit Methods

Dense/banded linear solvers.

abstols = 0.1 .^ (7:13)
reltols = 0.1 .^ (4:10)
setups = [
    Dict(:alg => KenCarp3()),
    Dict(:alg => KenCarp4()),
    Dict(:alg => KenCarp5()),
    Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:Band, jac_upper=1, jac_lower=1)),
    Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:Band, jac_upper=1, jac_lower=1)),
    Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Band, jac_upper=1, jac_lower=1))
]
labels = hcat(
    "KenCarp3",
    "KenCarp4",
    "KenCarp5",
    "ARKODE3",
    "ARKODE4",
    "ARKODE5",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    print_names=true, names=labels, numruns=5, error_estimate=:l2,
    save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));

plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Band Linsolve, Low Tolerances")

KenCarp3
KenCarp4
KenCarp5
ARKODE3
ARKODE4
ARKODE5
 37.088320 seconds (30.72 M allocations: 4.033 GiB, 5.18% gc time, 79.59% c
ompilation time)

Krylov linear solvers.

abstols = 0.1 .^ (7:13)
reltols = 0.1 .^ (4:10)
setups = [
    Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES())),
    Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES())),
    Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())),
    Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:GMRES)),
    Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:GMRES)),
    Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)),
]
labels = hcat(
    "KenCarp3",
    "KenCarp4",
    "KenCarp5",
    "ARKODE3",
    "ARKODE4",
    "ARKODE5",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    print_names=true, names=labels, numruns=5, error_estimate=:l2,
    save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));

plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances")

KenCarp3
KenCarp4
KenCarp5
ARKODE3
ARKODE4
ARKODE5
 49.934165 seconds (34.62 M allocations: 4.554 GiB, 4.69% gc time, 79.24% c
ompilation time)

Krylov solvers with preconditioners.

# Weighted diagonal preconditioner
import LinearAlgebra as LA
Base.@kwdef struct WeightedDiagonalPreconBuilder
    w::Float64
end

(builder::WeightedDiagonalPreconBuilder)(A, p) = (builder.w * LA.Diagonal(convert(AbstractMatrix, A)), LA.I)

# Incomplete LU factorization
using IncompleteLU
Base.@kwdef struct IncompleteLUPreconBuilder
    τ::Float64
end
function (builder::IncompleteLUPreconBuilder)(A, p)
    W = convert(AbstractMatrix, A)
    W = W isa SparseMatrixCSC ? W : sparse(W)
    Pl = ilu(W; τ = builder.τ)
    return Pl, LA.I
end

# Algebraic multigrid
using AlgebraicMultigrid
function algebraicmultigrid(A, p)
    W = convert(AbstractMatrix, A)
    W = W isa SparseMatrixCSC ? W : sparse(W)
    Pl = aspreconditioner(ruge_stuben(W))
    return Pl, LA.I
end
function algebraicmultigrid2(A, p)
    W = convert(AbstractMatrix, A)
    W = W isa SparseMatrixCSC ? W : sparse(W)
    Pl = AlgebraicMultigrid.aspreconditioner(AlgebraicMultigrid.ruge_stuben(W,
        presmoother = AlgebraicMultigrid.Jacobi(rand(size(W, 1))),
        postsmoother = AlgebraicMultigrid.Jacobi(rand(size(W, 1)))))
    return Pl, LA.I
end

# Aliases
inc_lu = IncompleteLUPreconBuilder(τ = 50.0)
w_diag = WeightedDiagonalPreconBuilder(w = 0.9)
amg = algebraicmultigrid
amg2 = algebraicmultigrid2

abstols = 0.1 .^ (8:13)
reltols = 0.1 .^ (5:10)
N = length(abstols)
setups = [
    # Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES(), :dts => 1e-2 * ones(N), :adaptive => true),
    # Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES(), :dts => 1e-2 * ones(N), :adaptive => true),
    # Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES(), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES(; precs = inc_lu), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES(; precs = inc_lu), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES(; precs = inc_lu), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES(; precs = w_diag), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES(; precs = w_diag), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES(; precs = w_diag), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES(; precs = amg), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES(; precs = amg), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES(; precs = amg), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),    
    Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES(; precs = amg2), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES(; precs = amg2), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES(; precs = amg2), concrete_jac = true), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES()), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES()), :dts => 1e-2 * ones(N), :adaptive => true),
    Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES()), :dts => 1e-2 * ones(N), :adaptive => true),
]
labels = hcat(
    # "KenCarp3 (Cholesky)",
    # "KenCarp4 (Cholesky)",
    # "KenCarp5 (Cholesky)",
    "KenCarp3 (ILU, τ = $(inc_lu.τ))",
    "KenCarp4 (ILU, τ = $(inc_lu.τ))",
    "KenCarp5 (ILU, τ = $(inc_lu.τ))",
    "KenCarp3 (Diagonal, w = $(w_diag.w))",
    "KenCarp4 (Diagonal, w = $(w_diag.w))",
    "KenCarp5 (Diagonal, w = $(w_diag.w))",
    "KenCarp3 (AMG)",
    "KenCarp4 (AMG)",
    "KenCarp5 (AMG)",
    "KenCarp3 (AMG-Jacobi)",
    "KenCarp4 (AMG-Jacobi)",
    "KenCarp5 (AMG-Jacobi)",
    "KenCarp3 (Identity)",
    "KenCarp4 (Identity)",
    "KenCarp5 (Identity)",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    print_names=true, names=labels, numruns=5, error_estimate=:l2,
    save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));

plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances")

KenCarp3 (ILU, τ = 50.0)
KenCarp4 (ILU, τ = 50.0)
KenCarp5 (ILU, τ = 50.0)
KenCarp3 (Diagonal, w = 0.9)
KenCarp4 (Diagonal, w = 0.9)
KenCarp5 (Diagonal, w = 0.9)
KenCarp3 (AMG)
KenCarp4 (AMG)
KenCarp5 (AMG)
KenCarp3 (AMG-Jacobi)
KenCarp4 (AMG-Jacobi)
KenCarp5 (AMG-Jacobi)
KenCarp3 (Identity)
KenCarp4 (Identity)
KenCarp5 (Identity)
205.803528 seconds (134.55 M allocations: 23.465 GiB, 4.65% gc time, 76.07%
 compilation time)

Exponential Integrators

abstols = 0.1 .^ (7:11) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (4:8)
multipliers = 0.5 .^ (0:4)
setups = [
    Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers),
    Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers),
    Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers),
]
labels = hcat(
    "ETDRK3 (caching)",
    "ETDRK4 (caching)",
    "HochOst4 (caching)",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    print_names=true, names=labels, numruns=5, error_estimate=:l2,
    save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));

plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances")

ETDRK3 (caching)
ETDRK4 (caching)
HochOst4 (caching)
121.044644 seconds (6.13 M allocations: 36.525 GiB, 1.21% gc time, 2.77% co
mpilation time)

Comparisons Between Families

abstols = 0.1 .^ (7:11)
reltols = 0.1 .^ (4:8)
multipliers = 0.5 .^ (0:4)
setups = [
    Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Band, jac_upper=1, jac_lower=1)),
    Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers),
    Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers),
]
labels = hcat(
    "ARKODE (Band linsolve)",
    "ETDRK3 (caching)",
    "ETDRK4 (caching)",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    print_names=true, names=labels, numruns=5, error_estimate=:l2,
    save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));

plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances")

ARKODE (Band linsolve)
ETDRK3 (caching)
ETDRK4 (caching)
101.362850 seconds (2.08 M allocations: 24.034 GiB, 0.84% gc time)

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/SimpleHandwrittenPDE","burgers_fdm_wpd.jmd")

Computer Information:

Julia Version 1.10.12
Commit d93beab124c (2026-08-15 10:29 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/SimpleHandwrittenPDE/Project.toml`
  [47edcb42] ADTypes v1.24.0
  [2169fc97] AlgebraicMultigrid v2.0.2
⌃ [b30e2e7b] ClassicalOrthogonalPolynomials v0.15.20
  [f3b72e0c] DiffEqDevTools v3.6.3
  [40713840] IncompleteLU v0.2.1
  [7f56f5a3] LSODA v1.2.0
⌃ [7ed4a6bd] LinearSolve v5.17.3
  [1dea7af3] OrdinaryDiffEq v7.8.1
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.9
⌃ [bbf590c4] OrdinaryDiffEqCore v4.17.2
  [e0540318] OrdinaryDiffEqExponentialRK v2.4.0
  [5960d6e9] OrdinaryDiffEqFIRK v2.8.7
  [d28bc4f8] OrdinaryDiffEqHighOrderRK v2.2.1
  [9f002381] OrdinaryDiffEqIMEXMultistep v2.3.0
⌅ [d4b830b4] OrdinaryDiffEqMultirate v2.7.0
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.7.3
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.4
⌃ [358294b1] OrdinaryDiffEqStabilizedRK v2.7.0
  [91a5bcdd] Plots v1.41.7
  [31c91b34] SciMLBenchmarks v0.2.1
⌃ [c0aeaf25] SciMLOperators v1.30.0
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  [0a514795] SparseMatrixColorings v0.4.28
  [9f78cca6] SummationByPartsOperators v0.5.97
  [c3572dad] Sundials v6.7.1
  [37e2e46d] LinearAlgebra
  [2f01184e] SparseArrays v1.10.0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`

And the full manifest:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/SimpleHandwrittenPDE/Manifest.toml`
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  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [781609d7] GMP_jll v6.2.1+6
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [3a97d323] MPFR_jll v4.2.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`