Burgers FDM Work-Precision Diagrams

using ApproxFun, OrdinaryDiffEq, Sundials, LinearSolve
using DiffEqDevTools
using LinearAlgebra
using Plots; gr()

Plots.GRBackend()

Here is the Burgers equation using FDM.

function lin_term(N, ϵ)
    dx = 1/(N + 1)
    d = -2 * ones(N) # main diagonal
    du = ones(N - 1) # off diagonal
    DiffEqArrayOperator((ϵ/dx^2) * diagm(-1 => du, 0 => d, 1 => du))
end

function nl_term(N)
    dx = 1/(N + 1)
    du = ones(N - 1) # super diagonal
    dl = -ones(N - 1) # lower diagonal
    D = (-1/(4*dx)) * diagm(-1 => dl, 1 => du)

    tmp = zeros(N)
    function (du,u,p,t)
        @. tmp = u^2
        mul!(du, D, tmp)
    end
end

# Construct the problem
function burgers(N, ϵ)
    f1 = lin_term(N, ϵ)
    f2 = nl_term(N)
    dx = 1 / (N + 1)
    xs = (1:N) * dx

    μ0 = 0.3; σ0 = 0.05
    f0 = x -> exp(-(x - μ0)^2 / (2 * σ0^2))
    u0 = f0.(xs)
    prob = SplitODEProblem(f1, f2, u0, (0.0, 1.0))
    xs, prob
end;

Reference solution using Vern9 is below:

xs, prob = burgers(512, 1e-3)
sol = solve(prob, Vern9(); abstol=1e-14, reltol=1e-14)
test_sol = TestSolution(sol);

tslices = [0.0 0.25 0.50 0.75 1.00]
ys = hcat((sol(t) for t in tslices)...)
labels = ["t = $t" for t in tslices]
plot(xs, ys, label=labels)

In-family comparisons

1.IMEX methods (dense 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(), :dts => 1e-3 * multipliers),
          Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers),
          Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers),
          Dict(:alg => SBDF2(), :dts => 1e-3 * multipliers)]
labels = ["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="IMEX methods, dense linsolve, low order")

IMEXEuler
CNAB2
CNLF2
SBDF2
1227.565700 seconds (164.63 M allocations: 9.914 GiB, 0.92% gc time, 0.96% 
compilation time)

1.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 => 1e-4 * multipliers),
          Dict(:alg => CNLF2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers),
          Dict(:alg => SBDF2(linsolve=KrylovJL_GMRES()), :dts => 1e-3 * multipliers)]
labels = ["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="IMEX methods, Krylov linsolve, low order")

IMEXEuler
CNAB2
CNLF2
SBDF2
1223.616236 seconds (151.26 M allocations: 9.016 GiB, 0.87% gc time, 0.27% 
compilation time)

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-3 * multipliers)]
          #Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-2 * multipliers)) matrix contains Inf or NaN
          #Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-2 * multipliers) matrix contains Inf or NaN
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, low order")

NorsettEuler (caching)
NorsettEuler (m=5)
NorsettEuler (m=20)
ETDRK2 (caching)
515.828176 seconds (11.44 M allocations: 126.821 GiB, 1.13% gc time, 1.28% 
compilation time)

Between 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 => 1e-4 * multipliers),
          Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers),
          Dict(:alg => ETDRK2(), :dts => 1e-3 * multipliers)]
labels = ["CNAB2 (dense linsolve)" "CNAB2 (Krylov linsolve)" "ETDRK2 (m=5)"]
@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 family, low orders")

CNAB2 (dense linsolve)
CNAB2 (Krylov linsolve)
ETDRK2 (m=5)
1162.177035 seconds (127.68 M allocations: 68.076 GiB, 0.93% gc time)

In-family comparisons

1.IMEX methods (dense linear solver)

abstols = 0.1 .^ (8:13)
reltols = 0.1 .^ (5:10)
setups = [Dict(:alg => KenCarp3()),
          Dict(:alg => KenCarp4()),
          Dict(:alg => KenCarp5()),
          Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:Dense)),
          Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:Dense)),
          Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Dense))]
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, dense linsolve, medium order")

KenCarp3
KenCarp4
KenCarp5
ARKODE3
ARKODE4
ARKODE5
14507.086587 seconds (965.89 M allocations: 61.079 GiB, 0.30% gc time, 0.09
% compilation time)

1.IMEX methods (Krylov linear solver)

abstols = 0.1 .^ (8:13)
reltols = 0.1 .^ (5:10)
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 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, medium order")

KenCarp3
KenCarp4
KenCarp5
ARKODE3
ARKODE4
ARKODE5
14523.073814 seconds (967.23 M allocations: 60.779 GiB, 0.29% gc time, 0.04
% compilation time)

2.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 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, medium order")

ETDRK3 (caching)
ETDRK3 (m=5)
ETDRK4 (caching)
ETDRK4 (m=5)
HochOst4 (caching)
HochOst4 (m=5)
1758.111377 seconds (20.96 M allocations: 465.691 GiB, 1.31% gc time, 0.72%
 compilation time)

Between family comparisons

abstols = 0.1 .^ (8:13)
reltols = 0.1 .^ (5:10)
multipliers = 0.5 .^ (0:5)
setups = [Dict(:alg => KenCarp4()),
          Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Dense)),
          Dict(:alg => KenCarp4(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("KenCarp4 (dense linsolve)", "ARKODE (dense linsolve)", "KenCarp4 (Krylov linsolve)",
              "ARKODE (Krylov linsolve)", "ETDRK3 (m=5)", "ETDRK4 (m=5)")
@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));#162s

plot(wp, label=labels, markershape=:auto, title="Between family, medium order")

KenCarp4 (dense linsolve)
ARKODE (dense linsolve)
KenCarp4 (Krylov linsolve)
ARKODE (Krylov linsolve)
ETDRK3 (m=5)
ETDRK4 (m=5)
4425.036799 seconds (266.98 M allocations: 24.498 GiB, 0.35% gc time, 0.00%
 compilation time)

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 1.9.3
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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  [aea7be01] PrecompileTools v1.2.0
  [21216c6a] Preferences v1.4.0
  [27ebfcd6] Primes v0.5.4
  [1fd47b50] QuadGK v2.8.2
  [74087812] Random123 v1.6.1
  [fb686558] RandomExtensions v0.4.3
  [e6cf234a] RandomNumbers v1.5.3
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  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [731186ca] RecursiveArrayTools v2.38.7
  [f2c3362d] RecursiveFactorization v0.2.20
  [189a3867] Reexport v1.2.2
  [05181044] RelocatableFolders v1.0.0
  [ae029012] Requires v1.3.0
  [ae5879a3] ResettableStacks v1.1.1
  [79098fc4] Rmath v0.7.1
  [47965b36] RootedTrees v2.19.2
  [7e49a35a] RuntimeGeneratedFunctions v0.5.12
  [fdea26ae] SIMD v3.4.5
  [94e857df] SIMDTypes v0.1.0
  [476501e8] SLEEFPirates v0.6.39
⌃ [0bca4576] SciMLBase v1.95.0
  [31c91b34] SciMLBenchmarks v0.1.3
  [e9a6253c] SciMLNLSolve v0.1.8
  [c0aeaf25] SciMLOperators v0.3.6
  [6c6a2e73] Scratch v1.2.0
  [efcf1570] Setfield v1.1.1
  [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.1.0
  [727e6d20] SimpleNonlinearSolve v0.1.19
  [699a6c99] SimpleTraits v0.9.4
  [ce78b400] SimpleUnPack v1.1.0
  [66db9d55] SnoopPrecompile v1.0.3
  [b85f4697] SoftGlobalScope v1.1.0
  [a2af1166] SortingAlgorithms v1.1.1
  [47a9eef4] SparseDiffTools v2.5.1
  [e56a9233] Sparspak v0.3.9
  [276daf66] SpecialFunctions v2.3.1
  [aedffcd0] Static v0.8.8
  [0d7ed370] StaticArrayInterface v1.4.1
  [90137ffa] StaticArrays v1.6.2
  [1e83bf80] StaticArraysCore v1.4.2
  [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.

Burgers FDM Work-Precision Diagrams

High tolerances

In-family comparisons

Between family comparisons

Low tolerances

In-family comparisons

Between family comparisons

Appendix