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Fitzhugh-Nagumo Work-Precision Diagrams

Fitzhugh-Nagumo

The purpose of this is to see how the errors scale on a standard nonlinear problem.

using OrdinaryDiffEq, ParameterizedFunctions, ODEInterface,
      ODEInterfaceDiffEq, LSODA, Sundials, DiffEqDevTools,
      StaticArrays, OrdinaryDiffEqSIMDRK
using OrdinaryDiffEqAdamsBashforthMoulton, OrdinaryDiffEqExtrapolation, OrdinaryDiffEqHighOrderRK, OrdinaryDiffEqLowOrderRK, OrdinaryDiffEqVerner
using SciMLLogging
using Plots;
gr()

f = @ode_def FitzhughNagumo begin
    dv = v - v^3/3 - w + l
    dw = τinv*(v + a - b*w)
end a b τinv l

p = SA[0.7, 0.8, 1 / 12.5, 0.5]
prob = ODEProblem{true, SciMLBase.FullSpecialize}(f, [1.0; 1.0], (0.0, 10.0), p)
probstatic = ODEProblem{false}(f, SA[1.0; 1.0], (0.0, 10.0), p)

abstols = 1.0 ./ 10.0 .^ (6:13)
reltols = 1.0 ./ 10.0 .^ (3:10);

sol = solve(prob, Vern7(), abstol = 1/10^14, reltol = 1/10^14)
sol2 = solve(probstatic, Vern7(), abstol = 1/10^14, reltol = 1/10^14)
probs = [prob, probstatic]
test_sol = [sol, sol2];
plot(sol)

Low Order

setups = [Dict(:alg=>DP5())
          #Dict(:alg=>ode45()) #fails
          Dict(:alg=>dopri5())
          Dict(:alg=>BS5())
          Dict(:alg=>Tsit5())
          Dict(:alg=>Vern6())
          Dict(:alg=>Tsit5(), :prob_choice => 2)
          Dict(:alg=>Vern6(), :prob_choice => 2)
          Dict(:alg=>MER5v2(), :prob_choice => 2)
          Dict(:alg=>MER6v2(), :prob_choice => 2)
          Dict(:alg=>RK6v4(), :prob_choice => 2)]
wp = WorkPrecisionSet(probs, abstols, reltols, setups; appxsol = test_sol,
    save_everystep = false, numruns = 100, maxiters = 1000)
plot(wp)

Interpolation

setups = [Dict(:alg=>DP5())
          #Dict(:alg=>ode45()) # fails
          Dict(:alg=>BS5())
          Dict(:alg=>Tsit5())
          Dict(:alg=>Vern6())
          Dict(:alg=>Tsit5(), :prob_choice => 2)
          Dict(:alg=>Vern6(), :prob_choice => 2)]
wp = WorkPrecisionSet(probs, abstols, reltols, setups; appxsol = test_sol, numruns = 100,
    maxiters = 10000, error_estimate = :L2, dense_errors = true)
plot(wp)

Higher Order

setups = [Dict(:alg=>DP8())
          Dict(:alg=>dop853())
          #Dict(:alg=>ode78()) # fails
          Dict(:alg=>Vern6())
          Dict(:alg=>Vern7())
          Dict(:alg=>Vern8())
          Dict(:alg=>Vern9())
          Dict(:alg=>Vern6(), :prob_choice => 2)
          Dict(:alg=>Vern7(), :prob_choice => 2)
          Dict(:alg=>Vern8(), :prob_choice => 2)
          Dict(:alg=>Vern9(), :prob_choice => 2)]
wp = WorkPrecisionSet(probs, abstols, reltols, setups; appxsol = test_sol,
    save_everystep = false, numruns = 100, maxiters = 1000)
plot(wp)

setups = [Dict(:alg=>DP8())
          Dict(:alg=>Vern7())
          Dict(:alg=>Vern7(), :prob_choice => 2)
          Dict(:alg=>CVODE_Adams())
          Dict(:alg=>ARKODE(Sundials.Explicit(), order = 6))
          Dict(:alg=>lsoda())
          Dict(:alg=>odex())
          Dict(:alg=>ddeabm())]
wp = WorkPrecisionSet(probs, abstols, reltols, setups; appxsol = test_sol,
    save_everystep = false, numruns = 100, maxiters = 1000)
plot(wp)

Interpolation

setups = [Dict(:alg=>DP8())
          #Dict(:alg=>ode78()) # fails
          Dict(:alg=>Vern6())
          Dict(:alg=>Vern7())
          Dict(:alg=>Vern8())
          Dict(:alg=>Vern9())
          Dict(:alg=>Vern6(), :prob_choice => 2)
          Dict(:alg=>Vern7(), :prob_choice => 2)
          Dict(:alg=>Vern8(), :prob_choice => 2)
          Dict(:alg=>Vern9(), :prob_choice => 2)]
wp = WorkPrecisionSet(probs, abstols, reltols, setups; appxsol = test_sol, numruns = 100,
    maxiters = 1000, error_estimate = :L2, dense_errors = true)
plot(wp)

Comparison with Non-RK methods

Now let's test Tsit5 and Vern9 against parallel extrapolation methods and an Adams-Bashforth-Moulton:

setups = [Dict(:alg=>Tsit5())
          Dict(:alg=>Vern9())
          Dict(:alg=>VCABM())
          Dict(:alg=>Vern9(), :prob_choice => 2)
          Dict(:alg=>VCABM(), :prob_choice => 2)
          Dict(:alg=>AitkenNeville(min_order = 1, max_order = 9, init_order = 4, threading = true))
          Dict(:alg=>ExtrapolationMidpointDeuflhard(
              min_order = 1, max_order = 9, init_order = 4, threading = true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(
              min_order = 2, max_order = 11, init_order = 4, threading = true))]
solnames = ["Tsit5", "Vern9", "VCABM", "Vern9 Static", "VCABM Static",
    "AitkenNeville", "Midpoint Deuflhard", "Midpoint Hairer Wanner"]
wp = WorkPrecisionSet(
    probs, abstols, reltols, setups; appxsol = test_sol, names = solnames,
    save_everystep = false, verbose = SciMLLogging.None(), numruns = 100)
plot(wp)

setups = [Dict(:alg=>ExtrapolationMidpointDeuflhard(
              min_order = 1, max_order = 9, init_order = 9, threading = false))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(
              min_order = 2, max_order = 11, init_order = 4, threading = false))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(
              min_order = 2, max_order = 11, init_order = 4, threading = true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(
              min_order = 2, max_order = 11, init_order = 4,
              sequence = :romberg, threading = true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(
              min_order = 2, max_order = 11, init_order = 4,
              sequence = :bulirsch, threading = true))]
solnames = ["Deuflhard", "No threads", "standard", "Romberg", "Bulirsch"]
wp = WorkPrecisionSet(
    probs, abstols, reltols, setups; appxsol = test_sol, names = solnames,
    save_everystep = false, verbose = SciMLLogging.None(), numruns = 100)
plot(wp)

setups = [Dict(:alg=>ExtrapolationMidpointHairerWanner(
              min_order = 2, max_order = 11, init_order = 10, threading = true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(
              min_order = 2, max_order = 11, init_order = 4, threading = true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(
              min_order = 5, max_order = 11, init_order = 10, threading = true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(
              min_order = 2, max_order = 15, init_order = 10, threading = true))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(
              min_order = 5, max_order = 7, init_order = 6, threading = true))]
solnames = ["1", "2", "3", "4", "5"]
wp = WorkPrecisionSet(
    probs, abstols, reltols, setups; appxsol = test_sol, names = solnames,
    save_everystep = false, verbose = SciMLLogging.None(), numruns = 100)
plot(wp)

Conclusion

As expected, the algorithms are all pretty matched on time for this problem. However, you can clearly see the OrdinaryDiffEq.jl algorithms solving to a much higher accuracy and still faster, especially when the interpolations are involved.

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/NonStiffODE","FitzhughNagumo_wpd.jmd")

Computer Information:

Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 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
  LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/NonStiffODE/Project.toml`
  [2b5f629d] DiffEqBase v7.21.1
  [f3b72e0c] DiffEqDevTools v3.6.3
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⌃ [bbf590c4] OrdinaryDiffEqCore v4.17.2
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  [37e2e46d] LinearAlgebra v1.11.0
  [9a3f8284] Random v1.11.0
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`

And the full manifest:

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  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.5.1+0
  [d2c73de3] GR_jll v0.73.27+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
  [61579ee1] Ghostscript_jll v9.55.1+0
  [7746bdde] Glib_jll v2.88.3+0
  [3b182d85] Graphite2_jll v1.3.16+0
  [2e76f6c2] HarfBuzz_jll v100.14004.0+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [aacddb02] JpegTurbo_jll v3.2.0+1
  [c1c5ebd0] LAME_jll v3.100.3+0
  [88015f11] LERC_jll v4.2.0+0
  [1d63c593] LLVMOpenMP_jll v23.1.1+0
  [aae0fff6] LSODA_jll v0.1.2+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 v2.42.0+0
  [89763e89] Libtiff_jll v4.7.3+0
  [38a345b3] Libuuid_jll v2.42.0+0
  [856f044c] MKL_jll v2025.2.0+0
  [c771fb93] ODEInterface_jll v0.0.2+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [656ef2d0] OpenBLAS32_jll v0.3.34+0
  [458c3c95] OpenSSL_jll v3.5.8+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
  [36c8627f] Pango_jll v1.58.2+0
  [30392449] Pixman_jll v0.46.4+0
  [c0090381] Qt6Base_jll v6.10.2+2
  [629bc702] Qt6Declarative_jll v6.10.2+2
  [ce943373] Qt6ShaderTools_jll v6.10.2+1
  [6de9746b] Qt6Svg_jll v6.10.2+0
  [e99dba38] Qt6Wayland_jll v6.10.2+1
  [f50d1b31] Rmath_jll v0.5.2+0
  [ca45d3f4] SuiteSparse32_jll v7.12.1+1
  [fb77eaff] Sundials_jll v7.5.0+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.4+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+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 v1.3.8+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
  [a51aa0fd] Xorg_libXi_jll v1.8.4+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+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 v2.47.0+2
  [c5fb5394] Xorg_xtrans_jll v1.6.0+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 v3.14.1+0
  [0ac62f75] libass_jll v0.17.5+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [8e53e030] libdrm_jll v2.4.134+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 v1.6.58+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
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`