KdV Pseudospectral Methods Work-Precision Diagrams
using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve
using DiffEqDevTools
using LinearAlgebra, RecursiveFactorization
using Plots; gr()Error: Failed to precompile Sundials [c3572dad-4567-51f8-b174-8c6c989267f4]
to "/cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f
953/compiled/v1.10/Sundials/jl_n8l7ee".Here is the Burgers equation using Fourier spectral methods.
S = Fourier()
n = 512
x = points(S, n)
D2 = Derivative(S,2)[1:n,1:n]
D = (Derivative(S) → S)[1:n,1:n]
T = ApproxFun.plan_transform(S, n)
Ti = ApproxFun.plan_itransform(S, n)
D3 = (Derivative(S,3) → S)[1:n,1:n]
û₀ = T*cos.(x)
δ=0.022
tmp = similar(û₀)
p = (D,T,Ti,similar(tmp),tmp)
function kdv(dû,û,p,t)
D,T,Ti,u,tmp = p
mul!(u,D,û)
mul!(tmp,Ti,u)
mul!(u,Ti,û)
@. tmp=u*tmp
mul!(u,T,tmp)
@.dû = 6*u
endkdv (generic function with 1 method)Reference solution using RadauIIA5 is below:
prob = SplitODEProblem(DiffEqArrayOperator(-D3), kdv, û₀, (0.0,5.0), p)
sol = solve(prob, RadauIIA5(autodiff=false); reltol=1e-14,abstol=1e-14)
test_sol = TestSolution(sol)
tslices=[0.0 1.0 2.0 3.0 5.0]
ys=hcat((Ti*sol(t) for t in tslices)...)
labels=["t=$t" for t in tslices]
plot(x,ys,label=labels)Error: Special factorization not handled in current default algorithmHigh tolerances
In-family comparisons
1.IMEX methods (diagonal linear solver)
abstols = 0.1 .^ (5:8)
reltols = 0.1 .^ (1:4)
multipliers = 0.5 .^ (0:3)
setups = [Dict(:alg => IMEXEuler(linsolve=diag_linsolve), :dts => 1e-5 * multipliers),
Dict(:alg => CNAB2(linsolve=diag_linsolve), :dts => 5e-7 * multipliers),
Dict(:alg => CNLF2(linsolve=diag_linsolve), :dts => 5e-7 * multipliers),
Dict(:alg => SBDF2(linsolve=diag_linsolve), :dts => 1e-5 * multipliers)]
labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"]
@time wp1 = WorkPrecisionSet(prob,abstols,reltols,setups;
print_names=true,names=labels,
numruns=5,seconds=5,
save_everystop=false,appxsol=test_sol,maxiters=Int(1e5));
plot(wp1,label=labels,markershape=:auto,title="IMEX methods, diagonal linsolve, low order")Error: UndefVarError: `diag_linsolve` not defined- ExpRK 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 => NorsettEuler(), :dts => 1e-3 * multipliers),
Dict(:alg => ETDRK2(), :dts => 1e-2 * multipliers)]
labels = hcat("NorsettEuler",
"ETDRK2 (caching)")
@time wp2 = WorkPrecisionSet(prob,abstols,reltols,setups;
print_names=true, names=labels,
numruns=5,
save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));
plot(wp2, label=labels, markershape=:auto, title="ExpRK methods, low order")Error: UndefVarError: `test_sol` not definedBetween family comparisons
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(linsolve=diag_linsolve), :dts => 5e-5 * multipliers),
Dict(:alg => ETDRK2(), :dts => 1e-2 * multipliers)]
labels = ["CNAB2 (diagonal linsolve)" "ETDRK2"]
@time wp3 = 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(wp3, label=labels, markershape=:auto, title="Between family, low orders")Error: UndefVarError: `diag_linsolve` not definedLow tolerances
In-family comparisons
1.IMEX methods (band linear solver)
abstols = 0.1 .^ (7:13)
reltols = 0.1 .^ (4:10)
setups = [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("ARKODE3", "ARKODE4", "ARKODE5")
@time wp4 = 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(wp4, label=labels, markershape=:auto, title="IMEX methods, band linsolve, medium order")Error: UndefVarError: `Sundials` not defined2.ExpRK methods
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)")#,"ETDRK4 (m=5)" "ETDRK3 (m=5)" "HochOst4 (m=5)")
@time wp5 = 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(wp5, label=labels, markershape=:auto, title="ExpRK methods, medium order")Error: UndefVarError: `test_sol` not definedBetween family comparisons
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 (nondiagonal linsolve)", "ETDRK3 ()", "ETDRK4 ()")
@time wp6 = 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(wp6, label=labels, markershape=:auto, title="Between family, medium order")Error: UndefVarError: `Sundials` not definedAppendix
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 1.10.8
Commit 4c16ff44be8 (2025-01-22 10:06 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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/SimpleHandwrittenPDE/Project.toml`
⌃ [28f2ccd6] ApproxFun v0.13.23
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⌃ [09606e27] ODEInterfaceDiffEq v3.13.2
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[31c91b34] SciMLBenchmarks v0.1.3
⌃ [c3572dad] Sundials v4.19.3
[37e2e46d] LinearAlgebra
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`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.And the full manifest:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/SimpleHandwrittenPDE/Manifest.toml`
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[44d3d7a6] Weave v0.10.12
⌅ [efce3f68] WoodburyMatrices v0.5.5
⌃ [ddb6d928] YAML v0.4.9
⌃ [c2297ded] ZMQ v1.2.2
⌃ [700de1a5] ZygoteRules v0.2.3
⌃ [6e34b625] Bzip2_jll v1.0.8+0
⌃ [83423d85] Cairo_jll v1.16.1+1
⌃ [2e619515] Expat_jll v2.5.0+0
⌅ [b22a6f82] FFMPEG_jll v4.4.2+2
⌃ [f5851436] FFTW_jll v3.3.10+0
⌃ [34b6f7d7] FastTransforms_jll v0.6.2+0
⌃ [a3f928ae] Fontconfig_jll v2.13.93+0
⌃ [d7e528f0] FreeType2_jll v2.13.1+0
⌃ [559328eb] FriBidi_jll v1.0.10+0
⌃ [0656b61e] GLFW_jll v3.3.8+0
⌅ [d2c73de3] GR_jll v0.72.9+1
[78b55507] Gettext_jll v0.21.0+0
⌃ [f8c6e375] Git_jll v2.36.1+2
⌃ [7746bdde] Glib_jll v2.74.0+2
⌃ [3b182d85] Graphite2_jll v1.3.14+0
⌅ [2e76f6c2] HarfBuzz_jll v2.8.1+1
⌅ [1d5cc7b8] IntelOpenMP_jll v2023.2.0+0
⌃ [aacddb02] JpegTurbo_jll v2.1.91+0
⌃ [c1c5ebd0] LAME_jll v3.100.1+0
⌅ [88015f11] LERC_jll v3.0.0+1
⌃ [1d63c593] LLVMOpenMP_jll v15.0.4+0
[aae0fff6] LSODA_jll v0.1.2+0
⌃ [dd4b983a] LZO_jll v2.10.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+1
⌃ [d4300ac3] Libgcrypt_jll v1.8.7+0
⌃ [7e76a0d4] Libglvnd_jll v1.6.0+0
⌃ [7add5ba3] Libgpg_error_jll v1.42.0+0
⌃ [94ce4f54] Libiconv_jll v1.17.0+0
⌃ [4b2f31a3] Libmount_jll v2.35.0+0
⌅ [89763e89] Libtiff_jll v4.5.1+1
⌃ [38a345b3] Libuuid_jll v2.36.0+0
⌅ [856f044c] MKL_jll v2023.2.0+0
[c771fb93] ODEInterface_jll v0.0.1+0
[e7412a2a] Ogg_jll v1.3.5+1
⌅ [458c3c95] OpenSSL_jll v1.1.22+0
⌃ [efe28fd5] OpenSpecFun_jll v0.5.5+0
⌃ [91d4177d] Opus_jll v1.3.2+0
⌃ [30392449] Pixman_jll v0.42.2+0
⌅ [c0090381] Qt6Base_jll v6.4.2+3
⌅ [f50d1b31] Rmath_jll v0.4.0+0
⌅ [fb77eaff] Sundials_jll v5.2.1+0
⌃ [a2964d1f] Wayland_jll v1.21.0+0
⌃ [2381bf8a] Wayland_protocols_jll v1.25.0+0
⌃ [02c8fc9c] XML2_jll v2.10.4+0
⌃ [aed1982a] XSLT_jll v1.1.34+0
⌃ [ffd25f8a] XZ_jll v5.4.4+0
⌃ [4f6342f7] Xorg_libX11_jll v1.8.6+0
⌃ [0c0b7dd1] Xorg_libXau_jll v1.0.11+0
⌃ [935fb764] Xorg_libXcursor_jll v1.2.0+4
⌃ [a3789734] Xorg_libXdmcp_jll v1.1.4+0
⌃ [1082639a] Xorg_libXext_jll v1.3.4+4
⌃ [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
⌃ [a51aa0fd] Xorg_libXi_jll v1.7.10+4
⌃ [d1454406] Xorg_libXinerama_jll v1.1.4+4
⌃ [ec84b674] Xorg_libXrandr_jll v1.5.2+4
⌃ [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
⌃ [14d82f49] Xorg_libpthread_stubs_jll v0.1.1+0
⌃ [c7cfdc94] Xorg_libxcb_jll v1.15.0+0
⌃ [cc61e674] Xorg_libxkbfile_jll v1.1.2+0
[12413925] Xorg_xcb_util_image_jll v0.4.0+1
[2def613f] Xorg_xcb_util_jll v0.4.0+1
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
⌃ [35661453] Xorg_xkbcomp_jll v1.4.6+0
[33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
⌃ [c5fb5394] Xorg_xtrans_jll v1.5.0+0
⌃ [8f1865be] ZeroMQ_jll v4.3.4+0
⌃ [3161d3a3] Zstd_jll v1.5.5+0
⌅ [214eeab7] fzf_jll v0.29.0+0
⌃ [a4ae2306] libaom_jll v3.4.0+0
⌃ [0ac62f75] libass_jll v0.15.1+0
⌃ [f638f0a6] libfdk_aac_jll v2.0.2+0
⌃ [b53b4c65] libpng_jll v1.6.38+0
⌃ [a9144af2] libsodium_jll v1.0.20+0
⌃ [f27f6e37] libvorbis_jll v1.3.7+1
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
⌃ [d8fb68d0] xkbcommon_jll v1.4.1+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 v0.6.3
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.9.0
[de0858da] Printf
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays
[10745b16] Statistics v1.9.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v1.0.2+0
[781609d7] GMP_jll v6.2.1+2
[deac9b47] LibCURL_jll v7.84.0+0
[29816b5a] LibSSH2_jll v1.10.2+0
[3a97d323] MPFR_jll v4.1.1+4
[c8ffd9c3] MbedTLS_jll v2.28.2+0
[14a3606d] MozillaCACerts_jll v2022.10.11
[4536629a] OpenBLAS_jll v0.3.21+4
[05823500] OpenLibm_jll v0.8.1+0
[efcefdf7] PCRE2_jll v10.42.0+0
[bea87d4a] SuiteSparse_jll v5.10.1+6
[83775a58] Zlib_jll v1.2.13+0
[8e850b90] libblastrampoline_jll v5.8.0+0
[8e850ede] nghttp2_jll v1.48.0+0
[3f19e933] p7zip_jll v17.4.0+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`
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.