Kuramoto-Sivashinsky Finite-Difference Method Work-Precision Diagrams

Problem Description

The Kuramoto-Sivashinsky partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 16,~T = 30$, with the following initial and periodic boundary conditions:

\[\begin{align} \partial_t u & = -\kappa u(t,x)\partial_x u(t,x) - \beta \partial_x^2 u(t,x) - \gamma \partial_x^4 u(t,x), \\ u(0,x) & = \cos\left(\frac{2\pi}{L} x\right), \\ u(t,-L) & = u(t,L) = 1. \end{align}\]

The spatial derivative operators are represented via finite difference approximations, incorporating the periodic boundary conditions on an equispaced grid of points $x_s \in [-L, L]$. The coefficients $\kappa = 1,~\beta = 1/2,~\gamma = 1/8$ are chosen to produce `interesting' behavior as seen in the reference solution below.

\[\begin{align} \frac{du}{dt} & = -\kappa u D_x u - \beta D_x^2 u - \gamma D_x^4 u, \\ u(0,x_s) & = \cos\left(\frac{2\pi}{L} x_s\right), \\ u(t,-L) & = u(t,L) = 1. \end{align}\]

Implementation

using OrdinaryDiffEq
using OrdinaryDiffEqBDF, OrdinaryDiffEqExponentialRK, OrdinaryDiffEqFIRK, OrdinaryDiffEqIMEXMultistep, OrdinaryDiffEqMultirate, OrdinaryDiffEqRosenbrock, OrdinaryDiffEqSDIRK, OrdinaryDiffEqStabilizedRK
using ADTypes: AutoFiniteDiff
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))

# Construct the problem
function kuramoto_sivashinsky(N, L, alpha)
    D1 = periodic_derivative_operator(derivative_order = 1, accuracy_order = 4,
                                      xmin = -L, xmax = L, N = N)
    D2 = periodic_derivative_operator(derivative_order = 2, accuracy_order = 4,
                                      xmin = -L, xmax = L, N = N)
    D4 = periodic_derivative_operator(derivative_order = 4, accuracy_order = 4,
                                      xmin = -L, xmax = L, N = N)

    x = SBP.grid(D1)
    u0 = @. cos(2π * x / L) # Initial condition
    p = (; D1, alpha)

    tspan = (0.0, 1.0)
    prob = SplitODEProblem(MatrixOperator(-p.alpha / 2 * (sparse(D2) + 1/4 * sparse(D4))),
                           nonlinear_convection!, 
                           u0, tspan, 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  # Number of grid points
L = 16.0  # Domain length
alpha = 30.0 # Time scaling factor
xs, prob = kuramoto_sivashinsky(N, L, alpha)
@time sol = solve(prob, RadauIIA5(autodiff=AutoFiniteDiff()); dt = 1e-4, abstol=1e-14, reltol=1e-14, adaptive=true)

test_sol = TestSolution(sol);

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

18.720262 seconds (17.88 M allocations: 1.273 GiB, 4.53% gc time, 62.16% c
ompilation time)

Work-Precision Diagrams

High Tolerances

Implicit-Explicit Methods

abstols = 0.1 .^ (3:6) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (3:6)
multipliers = 0.3 .^ (0:3)
setups = [
    Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers),
    Dict(:alg => CNAB2(), :dts => 1e-3 * multipliers),
    Dict(:alg => CNLF2(), :dts => 1e-3 * multipliers),
    Dict(:alg => SBDF2(), :dts => 1e-3 * 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(1e6));

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

IMEXEuler
CNAB2
CNLF2
SBDF2
112.700337 seconds (52.84 M allocations: 19.432 GiB, 9.52% gc time, 22.28% 
compilation time)

Exponential Integrators

abstols = 0.1 .^ (3:6) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (3:6)
multipliers = 0.3 .^ (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)
]
labels = hcat(
    "NorsettEuler (caching)", 
    "NorsettEuler (m=5)",
    "NorsettEuler (m=20)",
    "ETDRK2 (caching)", 
    "ETDRK2 (m=5)", 
    "ETDRK2 (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(1e6));

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)
283.458314 seconds (143.72 M allocations: 72.343 GiB, 10.14% gc time, 6.00%
 compilation time)

Stabilized Integrators

abstols = 0.1 .^ (3:6)
reltols = 0.1 .^ (3:6)

setups = [
    Dict(:alg => ROCK4()),
    Dict(:alg => TSRKC3()),
    Dict(:alg => RKC()),
    Dict(:alg => ROCK2()),
]
labels = hcat(
    "ROCK4",
    "TSRKC3",
    "RKC",
    "ROCK2",
)
@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="Stabilized Methods, High Tolerance")

ROCK4
TSRKC3
RKC
ROCK2
 18.912960 seconds (14.07 M allocations: 3.766 GiB, 9.48% gc time, 80.49% c
ompilation time)

Multirate Methods

The methods from OrdinaryDiffEqMultirate.jl consume the same SplitODEProblem: the first component (the linear term containing the fourth derivative operator) is advanced with m explicit substeps inside every macro step, while the nonlinear convection is evaluated only at the macro rate. Multirate integration is built for problems whose slow component is expensive per evaluation. Here the slow component costs two sparse matrix-vector products, no more than the fast one, so the substeps buy stability on a genuinely stiff operator without saving any slow work, and the comparison against CNAB2 and ETDRK2 measures what that costs relative to treating the stiff term implicitly or exactly.

The linear operator has spectral radius about 1.4e4. With macro steps up to 2e-3, the choice m = 32 puts the largest micro step at 2e-3 * 1.4e4 / 32, about 0.9, inside the explicit micro stability bound of 2 with a factor of two margin. A scan from m = 32 to m = 128 left every error essentially unchanged, so the macro error dominates and larger m is pure cost. The methods are run at fixed macro steps, matching the other fixed step methods in this file.

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. On this problem KenCarp4 gets the Krylov linear solver: its dense default spends minutes per solve even at loose tolerances here, the same cost problem the implicit block below documents. The adaptive sweep runs at 1e-4 to 1e-7 rather than this file's usual 1e-3 start because the Krylov KenCarp4 solve stalls into a singular factorization at 1e-3 on this problem; the fixed step methods are unaffected by the tolerance values.

abstols = 0.1 .^ (4:7) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (4:7)
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 = 32, order = 4), :adaptive => false, :dts => 2e-3 * multipliers),
    Dict(:alg => MRAB(k = 2, m = 32), :adaptive => false, :dts => 2e-3 * multipliers),
    Dict(:alg => MIS(m = 32), :adaptive => false, :dts => 2e-3 * multipliers),
    Dict(:alg => MRIGARKERK22a(m = 32), :adaptive => false, :dts => 2e-3 * multipliers),
    Dict(:alg => MRIGARKERK33a(m = 32), :adaptive => false, :dts => 2e-3 * multipliers),
    Dict(:alg => MRIGARKERK45a(m = 32), :adaptive => false, :dts => 2e-3 * multipliers),
    Dict(:alg => CNAB2(), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers),
    Dict(:alg => KenCarp4(linsolve = KrylovJL_GMRES())),
    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 (Krylov)",
    "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(1e6));

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

MREEF
MRAB
MIS
MRIGARKERK22a
MRIGARKERK33a
MRIGARKERK45a
CNAB2
ETDRK2
ETDRK4
KenCarp4 (Krylov)
Tsit5
Rodas5P
FBDF
NordsieckBDF
194.814828 seconds (79.26 M allocations: 38.623 GiB, 6.27% gc time, 27.00% 
compilation time)

The measured curves make this a much harder problem for the explicit multirate members than Burgers, and the cross family references decide it. FBDF dominates the figure outright: 7 to 9 milliseconds per solve across the whole sweep with errors from 6e-4 down to 1.4e-7, one to two orders of magnitude cheaper than any other method at matched error. Tsit5 on the joined problem runs at a stability capped, nearly constant 0.07 seconds while its error falls from 6e-6 to 1.1e-8 for free, and Rodas5P reaches 7.9e-8 in 0.16 seconds. Against that, the explicit multirate members are not competitive: MREEF needs 0.78 seconds for 1.1e-8, eleven times Tsit5's cost at the same error, and every decade carries the m = 32 micro stepping tax. Within the fixed step family the old reading still holds (MRIGARKERK33a and CNAB2 tie near 1e-4, the multirate members extend below the references' 2.4e-5 floor), MRAB is unusable here since the convection it freezes is not slow, and MRIGARKERK45a loses order at the loose end. With both components equally cheap there is nothing for the multirate structure to amortize; what these curves document is that the explicit members hold stability and order on a strongly stiff fast operator, and that adaptive single rate stiff solvers own this problem.

The implicit multirate members treat the slow component implicitly, so for them the split is used in the reverse orientation: the stiff linear term becomes the implicitly handled slow part and the nonlinear convection is micro stepped explicitly. In this orientation the methods act as IMEX integrators whose explicit part is subdivided m times per stage; the convection is not fast, the substeps resolve nothing, and m = 2 is kept at the smallest useful value. KenCarp3 and KenCarp4 on the original problem make the same implicit and explicit assignment of the two terms, which makes them the natural single rate references; they are given the Krylov linear solver here because their dense default costs minutes per solve on this problem at the tighter tolerances, as the low tolerance section below also shows. The cross family references from the figure above run here as well over this tolerance range. These methods are adaptive apart from ETDRK4, which keeps its fixed step convention.

Astiff = convert(AbstractMatrix, prob.f.f1.f)
p_stiff = (; D1 = prob.p.D1, alpha = prob.p.alpha, A = Astiff)
stiff_linear!(du, u, p, t) = mul!(du, p.A, u)
prob_swapped = SplitODEProblem(nonlinear_convection!, stiff_linear!,
                               prob.u0, prob.tspan, p_stiff)

abstols = 0.1 .^ (4:8)
reltols = 0.1 .^ (4:8)
setups = [
    Dict(:alg => MRIGARKIRK21a(m = 2, autodiff = AutoFiniteDiff())),
    Dict(:alg => MRIGARKESDIRK34a(m = 2, autodiff = AutoFiniteDiff())),
    Dict(:alg => MRIGARKESDIRK46a(m = 2, autodiff = AutoFiniteDiff())),
    Dict(:alg => KenCarp3(linsolve = KrylovJL_GMRES()), :prob_choice => 2),
    Dict(:alg => KenCarp4(linsolve = KrylovJL_GMRES()), :prob_choice => 2),
    Dict(:alg => ETDRK4(), :prob_choice => 2, :dts => 1e-3 * 0.5 .^ (0:4)),
    Dict(:alg => Tsit5(), :prob_choice => 3),
    Dict(:alg => Rodas5P(autodiff = AutoFiniteDiff()), :prob_choice => 3),
    Dict(:alg => FBDF(autodiff = AutoFiniteDiff()), :prob_choice => 3),
    Dict(:alg => NordsieckBDF(autodiff = AutoFiniteDiff()), :prob_choice => 3),
]
labels = hcat(
    "MRIGARKIRK21a",
    "MRIGARKESDIRK34a",
    "MRIGARKESDIRK46a",
    "KenCarp3 (Krylov)",
    "KenCarp4 (Krylov)",
    "ETDRK4",
    "Tsit5",
    "Rodas5P",
    "FBDF","NordsieckBDF",
)
@time wp = WorkPrecisionSet([prob_swapped, prob, prob_joined], abstols, reltols, setups;
    print_names=true, names=labels, numruns=5, error_estimate=:l2,
    save_everystep=false, appxsol=[test_sol, test_sol, test_sol], maxiters=Int(1e6));

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

MRIGARKIRK21a
MRIGARKESDIRK34a
MRIGARKESDIRK46a
KenCarp3 (Krylov)
KenCarp4 (Krylov)
ETDRK4
Tsit5
Rodas5P
FBDF
NordsieckBDF
233.768166 seconds (160.31 M allocations: 135.435 GiB, 25.31% gc time, 11.5
4% compilation time)

Within the split constrained comparison the reversed orientation works: MRIGARKIRK21a and MRIGARKESDIRK34a run clean order lines (the second order member from 3.8e-4 in 24 milliseconds to 3.1e-8 in 2.6 seconds, the third order member to 6.1e-10 at about 10 seconds), and against the KenCarp references making the same term assignment they come out 30 to 300 times faster at matched error, for example MRIGARKESDIRK34a at 3.3e-5 in 0.11 seconds against KenCarp4 at 4.4e-5 in 2.3 seconds. MRIGARKESDIRK46a is the cheapest member at mid accuracies (1.8e-6 in a quarter of a second) but its error stops responding at the tightest pairs, rising back toward 1e-6, so its tolerance is not a trustworthy control knob at that end. The cross family references cap the claim, though: FBDF on the joined problem reaches 1.2e-7 in under 15 milliseconds and Tsit5 reaches 9.3e-10 in 0.074 seconds, so at matched error the free choice single rate methods beat the best multirate member by one to two orders of magnitude. The reading these curves support is narrow: among methods constrained to treat this split implicitly and explicitly by term, the solve decoupled members are much cheaper than the IMEX family in this range; once that constraint is dropped, an adaptive BDF or a stability capped explicit method solves the problem faster than any of them.

Comparisons Between Families

abstols = 0.1 .^ (3:6)
reltols = 0.1 .^ (3:6)
multipliers = 0.3 .^ (0:3)
setups = [
    Dict(:alg => CNAB2(), :dts => 1e-3 * multipliers),
    Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers),
    Dict(:alg => ROCK4()),
    Dict(:alg => TSRKC3()),
]
labels = hcat(
    "CNAB2 (dense)",
    "CNAB2 (Krylov)", 
    "ETDRK2 (caching)",
    "ROCK4",
    "TSRKC3",
)
@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)
CNAB2 (Krylov)
ETDRK2 (caching)
ROCK4
TSRKC3
 66.092021 seconds (17.21 M allocations: 17.985 GiB, 11.02% gc time, 4.21% 
compilation time)

Low Tolerances

Implicit-Explicit Methods

Default and explicit Krylov linear solvers.

abstols = 0.1 .^ (7:11)
reltols = 0.1 .^ (7:11)
setups = [
    Dict(:alg => KenCarp3()),
    Dict(:alg => KenCarp4()),
    Dict(:alg => KenCarp5()),
    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 (default)",
    "KenCarp4 (default)",
    "KenCarp5 (default)",
    "KenCarp3 (Krylov)",
    "KenCarp4 (Krylov)",
    "KenCarp5 (Krylov)",
    "ARKODE3 (Krylov)",
    "ARKODE4 (Krylov)",
    "ARKODE5 (Krylov)",
)
@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, Low Tolerances")

KenCarp3 (default)
KenCarp4 (default)
KenCarp5 (default)
KenCarp3 (Krylov)
KenCarp4 (Krylov)
KenCarp5 (Krylov)
ARKODE3 (Krylov)
ARKODE4 (Krylov)
ARKODE5 (Krylov)
325.392564 seconds (253.19 M allocations: 46.132 GiB, 7.30% gc time, 16.30%
 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-4 * multipliers),
    Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers),
    Dict(:alg => HochOst4(), :dts => 1e-3 * 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(1e6));

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

ETDRK3 (caching)
ETDRK4 (caching)
HochOst4 (caching)
168.070463 seconds (34.78 M allocations: 54.655 GiB, 9.07% gc time, 3.88% c
ompilation 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=:GMRES)),
    Dict(:alg => ETDRK3(), :dts => 1e-4 * multipliers),
    Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers),
    Dict(:alg => ROCK4()),
    Dict(:alg => TSRKC3()),
]
labels = hcat(
    "ARKODE5 (Krylov)",
    "ETDRK3 (caching)",
    "ETDRK4 (caching)",
    "ROCK4",
    "TSRKC3",
)
@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(1e6));

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

ARKODE5 (Krylov)
ETDRK3 (caching)
ETDRK4 (caching)
ROCK4
TSRKC3
154.389894 seconds (77.11 M allocations: 54.057 GiB, 12.98% 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","ks_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 `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/SimpleHandwrittenPDE/Project.toml`
  [47edcb42] ADTypes v1.24.0
⌃ [2169fc97] AlgebraicMultigrid v2.0.1
  [b30e2e7b] ClassicalOrthogonalPolynomials v0.15.20
⌃ [f3b72e0c] DiffEqDevTools v3.4.0
  [40713840] IncompleteLU v0.2.1
⌃ [7f56f5a3] LSODA v1.1.0
⌃ [7ed4a6bd] LinearSolve v5.15.0
⌃ [1dea7af3] OrdinaryDiffEq v7.7.0
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.4
⌃ [bbf590c4] OrdinaryDiffEqCore v4.15.0
⌃ [e0540318] OrdinaryDiffEqExponentialRK v2.3.0
⌃ [5960d6e9] OrdinaryDiffEqFIRK v2.8.0
⌃ [d28bc4f8] OrdinaryDiffEqHighOrderRK v2.2.0
⌃ [9f002381] OrdinaryDiffEqIMEXMultistep v2.2.0
⌅ [d4b830b4] OrdinaryDiffEqMultirate v2.7.0
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.7.0
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.0
  [358294b1] OrdinaryDiffEqStabilizedRK v2.6.0
  [91a5bcdd] Plots v1.41.7
⌃ [31c91b34] SciMLBenchmarks v0.1.3
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⌃ [9f842d2f] SparseConnectivityTracer v1.2.2
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  [37e2e46d] LinearAlgebra
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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 `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/SimpleHandwrittenPDE/Manifest.toml`
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  [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 v17.6.1+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 -m`