Allen-Cahn Pseudospectral Methods Work-Precision Diagrams
using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve
using DiffEqDevTools
using LinearAlgebra
using Plots; gr()Plots.GRBackend()Here is the Allen-Cahn equation using Chebyshev spectral methods.
function cheb(N)
N==0 && return (0,1)
x = cos.(pi*(0:N)/N)
c = [2; ones(N-1,1); 2].*(-1).^(0:N)
X = hcat([x for i in 1:N+1]...)
dX = X-X'
D = (c*(1 ./c)')./(dX+I) # off-diagonal entries
D = D .- Diagonal(vec(sum(D,dims=2))) # diagonal entries
D,x
end
N = 128
ChebD2,x = cheb(N)
xx = x
x = x[2:N]
w = .53*x + .47*sin.(-1.5*pi*x) - x # use w = u-x to make BCs homogeneous
u = [1;w+x;-1]
ϵ=0.01
D2=ϵ*(ChebD2^2)[2:N, 2:N]
function allen_cahn(du,u,x,t)
@. du = (u + x) - (u + x)^3
endallen_cahn (generic function with 1 method)Reference solution using RadauIIA5 is below:
prob = SplitODEProblem(DiffEqArrayOperator(D2), allen_cahn, w, (0.0,5.0), x)
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(([1;x.+sol(t);-1] for t in tslices)...)
labels=["t=$t" for t in tslices]
plot(xx,ys,label=labels)Error: Non-concrete Jacobian not yet supported by RadauIIA5.High tolerances
In-family comparisons
1.IMEX methods (dense linear solver)
abstols = 0.1 .^ (5:8)
reltols = 0.1 .^ (1:4)
multipliers = 0.5 .^ (0:3)
setups = [Dict(:alg => IMEXEuler(), :dts => 1e-3 * multipliers),
Dict(:alg => CNAB2(), :dts => 5e-3 * multipliers),
Dict(:alg => CNLF2(), :dts => 5e-4 * multipliers),
Dict(:alg => SBDF2(), :dts => 1e-3 * 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, dense linsolve, low order")Error: UndefVarError: `test_sol` not defined1.IMEX methods (Krylov linear solver)
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(linsolve=KrylovJL_GMRES()), :dts => 1e-3 * multipliers),
Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 5e-3 * multipliers),
Dict(:alg => CNLF2(linsolve=KrylovJL_GMRES()), :dts => 5e-4 * multipliers),
Dict(:alg => SBDF2(linsolve=KrylovJL_GMRES()), :dts => 1e-3 * multipliers)]
labels = ["IMEXEuler" "CNAB2" "CNLF2" "SBDF2"]
@time wp1 = 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(wp1, label=labels, markershape=:auto, title="IMEX methods, Krylov linsolve, low order")Error: UndefVarError: `test_sol` 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 => NorsettEuler(krylov=true, m=5), :dts => 1e-3 * multipliers),
Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-3 * multipliers),
Dict(:alg => ETDRK2(), :dts => 1e-2 * multipliers),
Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-2 * multipliers),
Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-2 * multipliers)]
labels = hcat("NorsettEuler (caching)", "NorsettEuler (m=5)", "NorsettEuler (m=20)",
"ETDRK2 (caching)", "ETDRK2 (m=5)", "ETDRK2 (m=20)")
@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(), :dts => 5e-3 * multipliers),
Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 5e-3 * multipliers),
Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-2 * multipliers)]
labels = ["CNAB2 (dense linsolve)" "CNAB2 (Krylov linsolve)" "ETDRK2 (m=5)"]
@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: `test_sol` not definedLow tolerances
In-family comparisons
1.IMEX methods (dense/band linear solver)
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 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, dense/band linsolve, medium order")Error: UndefVarError: `test_sol` not defined1.IMEX methods (krylov linear solver)
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 = ["KenCarp3" "KenCarp4" "KenCarp5" "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, Krylov linsolve, medium order")Error: UndefVarError: `test_sol` 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 => ETDRK3(krylov=true, m=5), :dts => 1e-2 * multipliers),
Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers),
Dict(:alg => ETDRK4(krylov=true, m=5), :dts => 1e-2 * multipliers),
Dict(:alg => HochOst4(), :dts => 1e-2 * multipliers),
Dict(:alg => HochOst4(krylov=true, m=5), :dts => 1e-2 * multipliers)]
labels = hcat("ETDRK3 (caching)", "ETDRK3 (m=5)", "ETDRK4 (caching)",
"ETDRK4 (m=5)", "HochOst4 (caching)", "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 => KenCarp5()),
Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Dense)),
Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())),
Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)),
Dict(:alg => ETDRK3(krylov=true, m=5), :dts => 1e-2 * multipliers),
Dict(:alg => ETDRK4(krylov=true, m=5), :dts => 1e-2 * multipliers)]
labels = hcat("KenCarp5 (dense linsolve)", "ARKODE (dense linsolve)", "KenCarp5 (Krylov linsolve)",
"ARKODE (Krylov linsolve)", "ETDRK3 (m=5)", "ETDRK4 (m=5)")
@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: `test_sol` 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","allen_cahn_spectral_wpd.jmd")Computer Information:
Julia Version 1.9.3
Commit bed2cd540a1 (2023-08-24 14:43 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-14.0.6 (ORCJIT, znver2)
Threads: 128 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`
[28f2ccd6] ApproxFun v0.13.23
[f3b72e0c] DiffEqDevTools v2.35.0
[5b8099bc] DomainSets v0.6.7
[7f56f5a3] LSODA v0.7.4
[7ed4a6bd] LinearSolve v2.5.1
[94925ecb] MethodOfLines v0.9.5
⌃ [961ee093] ModelingToolkit v8.66.0
[09606e27] ODEInterfaceDiffEq v3.13.2
[1dea7af3] OrdinaryDiffEq v6.55.0
[91a5bcdd] Plots v1.39.0
[31c91b34] SciMLBenchmarks v0.1.3
[c3572dad] Sundials v4.19.3
[37e2e46d] LinearAlgebra
Info Packages marked with ⌃ have new versions available and may be upgradable.
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-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/SimpleHandwrittenPDE/Manifest.toml`
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[82ae8749] StatsAPI v1.6.0
[2913bbd2] StatsBase v0.34.0
[4c63d2b9] StatsFuns v1.3.0
[7792a7ef] StrideArraysCore v0.4.17
[69024149] StringEncodings v0.3.7
[c3572dad] Sundials v4.19.3
[2efcf032] SymbolicIndexingInterface v0.2.2
⌃ [d1185830] SymbolicUtils v1.2.0
[0c5d862f] Symbolics v5.5.1
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.10.1
[62fd8b95] TensorCore v0.1.1
[8ea1fca8] TermInterface v0.3.3
[8290d209] ThreadingUtilities v0.5.2
[a759f4b9] TimerOutputs v0.5.23
[c751599d] ToeplitzMatrices v0.8.2
[0796e94c] Tokenize v0.5.25
[3bb67fe8] TranscodingStreams v0.9.13
[a2a6695c] TreeViews v0.3.0
[d5829a12] TriangularSolve v0.1.19
[410a4b4d] Tricks v0.1.7
[781d530d] TruncatedStacktraces v1.4.0
[5c2747f8] URIs v1.5.0
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
[1986cc42] Unitful v1.17.0
[45397f5d] UnitfulLatexify v1.6.3
[a7c27f48] Unityper v0.1.5
[41fe7b60] Unzip v0.2.0
[3d5dd08c] VectorizationBase v0.21.64
[81def892] VersionParsing v1.3.0
[19fa3120] VertexSafeGraphs v0.2.0
[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, 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.