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Korteweg-De Vries Equation Pseudospectral Methods Work-Precision Diagrams

using OrdinaryDiffEq
using DiffEqDevTools
using SciMLOperators
using LinearSolve
using LinearAlgebra
using SparseArrays
using Sundials
using SummationByPartsOperators
const SBP = SummationByPartsOperators
using Plots
gr()
Plots.GRBackend()

Problem Description

The Kortwrieg de Vries (KdV) partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 16,~T = 5$, with the following initial and periodic boundary conditions:

\[\begin{align} \partial_t u(t,x) & = -6u(t,x)\partial_x u(t,x) -\partial_x^3 u(t,x), \\ u(0,x) & = \cos\left(\frac{\pi x}{L}\right), \\ u(t,-L) & = u(t,L) = -1. \end{align}\]

The spatial derivative operators are represented via Fourier pseudospectral approximations. Here, the domain is discretized by projecting on an equispaced grid of points $x_s \in [-L, L]$; the solution is approximated on this grid via linear combinations of sinusoidal functions in space.

\[\begin{align} \frac{du}{dt} & = -6uD_x u -D_x^3 u, \\ u(0,x_s) & = \cos\left(\frac{\pi x_s}{L}\right), \\ u(t,-L) & = u(t,L) = -1. \end{align}\]

Implementation

nonlinear_convection!(du, u, p, t) = du .= (-6 * p.alpha / 3) * (u .* (p.D1 * u) + p.D1 * (u .^ 2))

function korteweg_de_vries(N, L, alpha)
    D1 = fourier_derivative_operator(xmin = -L, xmax = L, N = N)
    D3 = D1^3 # 3rd derivative operator
    x = SBP.grid(D1)
    u0 = @. cos(π * x / L) # Initial condition
    p = (; D1, alpha)

    tspan = (0.0, 1.0)
    prob = SplitODEProblem(MatrixOperator(-p.alpha * Matrix(D3)),
                           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.

L = 16.0 # Domain length
n = 128 # Number of Chebyshev points
alpha = 5.0 # Time scaling factor
xs, prob = korteweg_de_vries(n, L, alpha)

@time sol = solve(prob, AutoVern7(RadauIIA5(autodiff=false)); 
                  dt = 1e-4, reltol = 1e-12, abstol = 1e-12);

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

tslices = LinRange(prob.tspan..., 50)
ys = mapreduce(sol, hcat, tslices)
plt = heatmap(xs, tslices, ys', xlabel = "x", ylabel = "t")
15.906107 seconds (22.15 M allocations: 2.084 GiB, 8.45% gc time, 70.04% c
ompilation time)

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
115.892217 seconds (57.12 M allocations: 25.408 GiB, 2.28% gc time, 8.65% c
ompilation 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-4 * multipliers),
    Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-4 * multipliers),
    Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-4 * 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(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)
313.177190 seconds (96.87 M allocations: 102.938 GiB, 5.15% gc time, 2.66% 
compilation time)

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)
 77.621096 seconds (32.21 M allocations: 26.481 GiB, 7.37% gc time, 2.60% c
ompilation time)

Low Tolerances

Implicit-Explicit Methods

abstols = 0.1 .^ (8:10)
reltols = 0.1 .^ (5:7)
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",
    "KenCarp4",
    # "KenCarp5",
    "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, Krylov Linsolve, Low Tolerances")
KenCarp3
KenCarp4
KenCarp3 (Krylov)
KenCarp4 (Krylov)
ARKODE3 (Krylov)
ARKODE4 (Krylov)
ARKODE5 (Krylov)
527.000657 seconds (116.48 M allocations: 18.520 GiB, 0.44% gc time, 2.18% 
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-3 * 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(1e5));

plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances")
ETDRK3 (caching)
ETDRK4 (caching)
HochOst4 (caching)
178.124678 seconds (23.51 M allocations: 31.310 GiB, 2.72% gc time, 3.46% 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-3 * multipliers),
    Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers),
]
labels = hcat(
    "ARKODE (Krylov 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 (Krylov linsolve)
ETDRK3 (caching)
ETDRK4 (caching)
162.017249 seconds (66.71 M allocations: 30.330 GiB, 2.62% gc time, 0.00% c
ompilation 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","kdv_spectral_wpd.jmd")

Computer Information:

Julia Version 1.10.10
Commit 95f30e51f41 (2025-06-27 09:51 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: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
  JULIA_CPU_THREADS = 128
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953

Package Information:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/SimpleHandwrittenPDE/Project.toml`
  [b30e2e7b] ClassicalOrthogonalPolynomials v0.15.3
  [f3b72e0c] DiffEqDevTools v2.48.0
  [7f56f5a3] LSODA v0.7.5
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  [1dea7af3] OrdinaryDiffEq v6.101.0
  [91a5bcdd] Plots v1.40.17
  [31c91b34] SciMLBenchmarks v0.1.3
  [c0aeaf25] SciMLOperators v1.4.0
  [9f78cca6] SummationByPartsOperators v0.5.81
  [c3572dad] Sundials v4.28.0
  [37e2e46d] LinearAlgebra
  [2f01184e] SparseArrays v1.10.0

And the full manifest:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/SimpleHandwrittenPDE/Manifest.toml`
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  [ae029012] Requires v1.3.1
  [ae5879a3] ResettableStacks v1.1.1
  [79098fc4] Rmath v0.8.0
  [47965b36] RootedTrees v2.23.1
  [7e49a35a] RuntimeGeneratedFunctions v0.5.15
  [94e857df] SIMDTypes v0.1.0
  [476501e8] SLEEFPirates v0.6.43
  [0bca4576] SciMLBase v2.107.0
  [31c91b34] SciMLBenchmarks v0.1.3
  [19f34311] SciMLJacobianOperators v0.1.8
  [c0aeaf25] SciMLOperators v1.4.0
  [53ae85a6] SciMLStructures v1.7.0
  [6c6a2e73] Scratch v1.3.0
  [f8ebbe35] SemiseparableMatrices v0.4.0
  [efcf1570] Setfield v1.1.2
  [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.2.0
  [727e6d20] SimpleNonlinearSolve v2.7.0
  [ce78b400] SimpleUnPack v1.1.0
  [b85f4697] SoftGlobalScope v1.1.0
  [a2af1166] SortingAlgorithms v1.2.1
  [0a514795] SparseMatrixColorings v0.4.21
  [276daf66] SpecialFunctions v2.5.1
  [860ef19b] StableRNGs v1.0.3
  [aedffcd0] Static v1.2.0
  [0d7ed370] StaticArrayInterface v1.8.0
  [90137ffa] StaticArrays v1.9.14
  [1e83bf80] StaticArraysCore v1.4.3
  [82ae8749] StatsAPI v1.7.1
  [2913bbd2] StatsBase v0.34.5
  [4c63d2b9] StatsFuns v1.5.0
  [7792a7ef] StrideArraysCore v0.5.7
  [69024149] StringEncodings v0.3.7
  [892a3eda] StringManipulation v0.4.1
  [09ab397b] StructArrays v0.7.1
  [9f78cca6] SummationByPartsOperators v0.5.81
  [c3572dad] Sundials v4.28.0
  [2efcf032] SymbolicIndexingInterface v0.3.42
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.1
  [62fd8b95] TensorCore v0.1.1
  [8290d209] ThreadingUtilities v0.5.5
  [a759f4b9] TimerOutputs v0.5.29
  [c751599d] ToeplitzMatrices v0.8.5
  [3bb67fe8] TranscodingStreams v0.11.3
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.6.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.24.0
  [45397f5d] UnitfulLatexify v1.7.0
  [9602ed7d] Unrolled v0.1.5
  [41fe7b60] Unzip v0.2.0
  [3d5dd08c] VectorizationBase v0.21.71
  [81def892] VersionParsing v1.3.0
  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.14
  [c2297ded] ZMQ v1.4.1
  [6e34b625] Bzip2_jll v1.0.9+0
  [83423d85] Cairo_jll v1.18.5+0
  [ee1fde0b] Dbus_jll v1.16.2+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.6.5+0
  [b22a6f82] FFMPEG_jll v7.1.1+0
  [f5851436] FFTW_jll v3.3.11+0
  [34b6f7d7] FastTransforms_jll v0.6.3+0
  [a3f928ae] Fontconfig_jll v2.16.0+0
  [d7e528f0] FreeType2_jll v2.13.4+0
  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.4.0+2
  [d2c73de3] GR_jll v0.73.17+0
  [b0724c58] GettextRuntime_jll v0.22.4+0
  [f8c6e375] Git_jll v2.50.1+0
  [7746bdde] Glib_jll v2.84.3+0
  [3b182d85] Graphite2_jll v1.3.15+0
  [2e76f6c2] HarfBuzz_jll v8.5.1+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [aacddb02] JpegTurbo_jll v3.1.1+0
  [c1c5ebd0] LAME_jll v3.100.3+0
  [88015f11] LERC_jll v4.0.1+0
  [1d63c593] LLVMOpenMP_jll v18.1.8+0
  [aae0fff6] LSODA_jll v0.1.2+0
  [dd4b983a] LZO_jll v2.10.3+0
  [e9f186c6] Libffi_jll v3.4.7+0
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.41.0+0
  [89763e89] Libtiff_jll v4.7.1+0
  [38a345b3] Libuuid_jll v2.41.0+0
  [856f044c] MKL_jll v2025.2.0+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [9bd350c2] OpenSSH_jll v10.0.1+0
  [458c3c95] OpenSSL_jll v3.5.1+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.5.2+0
  [36c8627f] Pango_jll v1.56.3+0
⌅ [30392449] Pixman_jll v0.44.2+0
  [c0090381] Qt6Base_jll v6.8.2+1
  [629bc702] Qt6Declarative_jll v6.8.2+1
  [ce943373] Qt6ShaderTools_jll v6.8.2+1
  [e99dba38] Qt6Wayland_jll v6.8.2+1
  [f50d1b31] Rmath_jll v0.5.1+0
⌅ [fb77eaff] Sundials_jll v5.2.2+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.1+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.12+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.7+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.1+0
  [a51aa0fd] Xorg_libXi_jll v1.8.3+0
  [d1454406] Xorg_libXinerama_jll v1.1.6+0
  [ec84b674] Xorg_libXrandr_jll v1.5.5+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.3+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.5+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.44.0+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [8f1865be] ZeroMQ_jll v4.3.6+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
  [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.12.1+0
  [0ac62f75] libass_jll v0.17.4+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.50+0
  [a9144af2] libsodium_jll v1.0.21+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.0.0+0
  [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.9.2+0
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [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 ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`