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Enright-Pryce Work-Precision Diagrams

Enright-Pryce Suite

The purpose of this benchmark is to test explicit Runge-Kutta methods on the Enright-Pryce test suite, a collection of non-stiff ODE problems designed for testing numerical integrators.

using OrdinaryDiffEq, ParameterizedFunctions, ODEInterface,
      ODEInterfaceDiffEq, LSODA, Sundials, DiffEqDevTools,
      StaticArrays, OrdinaryDiffEqSIMDRK
using Plots
gr()

# Load the problems from the Enright-Pryce suite
include(joinpath(@__DIR__, "enright_pryce.jl"))

abstols = 1.0 ./ 10.0 .^ (6:13)
reltols = 1.0 ./ 10.0 .^ (3:10)
8-element Vector{Float64}:
 0.001
 0.0001
 1.0e-5
 1.0e-6
 1.0e-7
 1.0e-8
 1.0e-9
 1.0e-10

SIMD Algorithms

simdrkalgs = [
    Dict(:alg=>MER5v2()),
    Dict(:alg=>MER6v2()),
    Dict(:alg=>RK6v4())
]
3-element Vector{Dict{Symbol}}:
 Dict(:alg => MER5v2(stage_limiter! = trivial_limiter!, step_limiter! = tri
vial_limiter!, thread = static(false)))
 Dict(:alg => MER6v2(stage_limiter! = trivial_limiter!, step_limiter! = tri
vial_limiter!, thread = static(false)))
 Dict(:alg => RK6v4(stage_limiter! = trivial_limiter!, step_limiter! = triv
ial_limiter!, thread = static(false)))

Low Order Setup

setups = [
    Dict(:alg=>Tsit5()),
    Dict(:alg=>Vern6()),
    Dict(:alg=>Vern7()),
    Dict(:alg=>Vern9()),
    simdrkalgs...
]
7-element Vector{Dict{Symbol}}:
 Dict(:alg => OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_l
imiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}(Ordina
ryDiffEqCore.trivial_limiter!, OrdinaryDiffEqCore.trivial_limiter!, static(
false)))
 Dict{Symbol, OrdinaryDiffEqVerner.Vern6{typeof(OrdinaryDiffEqCore.trivial_
limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}}(:alg
 => OrdinaryDiffEqVerner.Vern6{typeof(OrdinaryDiffEqCore.trivial_limiter!),
 typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}(OrdinaryDiffEqC
ore.trivial_limiter!, OrdinaryDiffEqCore.trivial_limiter!, static(false), t
rue))
 Dict{Symbol, OrdinaryDiffEqVerner.Vern7{typeof(OrdinaryDiffEqCore.trivial_
limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}}(:alg
 => OrdinaryDiffEqVerner.Vern7{typeof(OrdinaryDiffEqCore.trivial_limiter!),
 typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}(OrdinaryDiffEqC
ore.trivial_limiter!, OrdinaryDiffEqCore.trivial_limiter!, static(false), t
rue))
 Dict{Symbol, OrdinaryDiffEqVerner.Vern9{typeof(OrdinaryDiffEqCore.trivial_
limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}}(:alg
 => OrdinaryDiffEqVerner.Vern9{typeof(OrdinaryDiffEqCore.trivial_limiter!),
 typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}(OrdinaryDiffEqC
ore.trivial_limiter!, OrdinaryDiffEqCore.trivial_limiter!, static(false), t
rue))
 Dict(:alg => MER5v2(stage_limiter! = trivial_limiter!, step_limiter! = tri
vial_limiter!, thread = static(false)))
 Dict(:alg => MER6v2(stage_limiter! = trivial_limiter!, step_limiter! = tri
vial_limiter!, thread = static(false)))
 Dict(:alg => RK6v4(stage_limiter! = trivial_limiter!, step_limiter! = triv
ial_limiter!, thread = static(false)))

Non-Stiff NA Problems

NA1

prob = NA_PROBLEMS[1]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NA1")

NA2

prob = NA_PROBLEMS[2]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NA2")

NA4

prob = NA_PROBLEMS[4]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NA4")

NA5

prob = NA_PROBLEMS[5]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NA5")

Non-Stiff NB Problems

NB1

prob = NB_PROBLEMS[1]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NB1")

NB2

prob = NB_PROBLEMS[2]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NB2")

NB3

prob = NB_PROBLEMS[3]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NB3")

NB5

prob = NB_PROBLEMS[5]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NB5")

Non-Stiff NC Problems

NC1

prob = NC_PROBLEMS[1]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NC1")

NC2

prob = NC_PROBLEMS[2]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NC2")

NC3

prob = NC_PROBLEMS[3]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NC3")

NC4

prob = NC_PROBLEMS[4]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NC4")

NC5

prob = NC_PROBLEMS[5]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="NC5")
Error: BoundsError: attempt to access 4-element Vector{SciMLBase.ODEProblem
{uType, Tuple{Float64, Float64}, false, ModelingToolkit.MTKParameters{Stati
cArraysCore.SizedVector{0, Float64, Vector{Float64}}, Vector{Float64}, Tupl
e{}, Tuple{}, Tuple{}, Tuple{}}, F, Base.Pairs{Symbol, Union{}, Tuple{}, @N
amedTuple{}}, SciMLBase.StandardODEProblem} where {uType, F}} at index [5]

Non-Stiff ND Problems

ND1

prob = ND_PROBLEMS[1]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="ND1")
Error: StackOverflowError:

ND2

prob = ND_PROBLEMS[2]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="ND2")
Error: StackOverflowError:

ND3

prob = ND_PROBLEMS[3]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="ND3")
Error: StackOverflowError:

ND4

prob = ND_PROBLEMS[4]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="ND4")
Error: StackOverflowError:

ND5

prob = ND_PROBLEMS[5]
test_sol = solve(prob, Vern9(), abstol=1/10^14, reltol=1/10^14)
wp = WorkPrecisionSet(prob, abstols, reltols, setups; appxsol=test_sol, save_everystep=false, numruns=100)
plot(wp; title="ND5")
Error: StackOverflowError:

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","EnrightPryce_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/NonStiffODE/Project.toml`
⌃ [2b5f629d] DiffEqBase v6.184.0
  [f3b72e0c] DiffEqDevTools v2.48.0
  [7f56f5a3] LSODA v0.7.5
  [961ee093] ModelingToolkit v10.21.0
  [54ca160b] ODEInterface v0.5.0
  [09606e27] ODEInterfaceDiffEq v3.13.4
  [1dea7af3] OrdinaryDiffEq v6.102.0
  [dc97f408] OrdinaryDiffEqSIMDRK v1.1.0
  [65888b18] ParameterizedFunctions v5.19.0
  [91a5bcdd] Plots v1.40.19
  [31c91b34] SciMLBenchmarks v0.1.3
⌃ [90137ffa] StaticArrays v1.9.14
  [c3572dad] Sundials v4.28.0
  [0c5d862f] Symbolics v6.52.0
  [37e2e46d] LinearAlgebra
  [9a3f8284] Random
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/NonStiffODE/Manifest.toml`
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  [efcf1570] Setfield v1.1.2
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  [727e6d20] SimpleNonlinearSolve v2.7.0
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  [a2af1166] SortingAlgorithms v1.2.2
  [0a514795] SparseMatrixColorings v0.4.21
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  [860ef19b] StableRNGs v1.0.3
  [aedffcd0] Static v1.2.0
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⌃ [90137ffa] StaticArrays v1.9.14
  [1e83bf80] StaticArraysCore v1.4.3
  [82ae8749] StatsAPI v1.7.1
  [2913bbd2] StatsBase v0.34.6
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  [bd369af6] Tables v1.12.1
  [ed4db957] TaskLocalValues v0.1.3
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  [8ea1fca8] TermInterface v2.0.0
  [1c621080] TestItems v1.0.0
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  [a759f4b9] TimerOutputs v0.5.29
  [3bb67fe8] TranscodingStreams v0.11.3
  [410a4b4d] Tricks v0.1.12
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.6.1
  [3a884ed6] UnPack v1.0.2
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  [1986cc42] Unitful v1.24.0
  [45397f5d] UnitfulLatexify v1.7.0
  [a7c27f48] Unityper v0.1.6
  [41fe7b60] Unzip v0.2.0
⌃ [3d5dd08c] VectorizationBase v0.21.71
  [81def892] VersionParsing v1.3.0
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  [ddb6d928] YAML v0.4.14
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  [ee1fde0b] Dbus_jll v1.16.2+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
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  [b22a6f82] FFMPEG_jll v7.1.1+0
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  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.4.0+2
  [d2c73de3] GR_jll v0.73.17+0
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  [020c3dae] Git_LFS_jll v3.7.0+0
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  [2e76f6c2] HarfBuzz_jll v8.5.1+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
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  [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 v2.41.1+0
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  [38a345b3] Libuuid_jll v2.41.1+0
  [856f044c] MKL_jll v2025.2.0+0
  [c771fb93] ODEInterface_jll v0.0.2+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [9bd350c2] OpenSSH_jll v10.0.1+0
  [458c3c95] OpenSSL_jll v3.5.2+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
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  [2db6ffa8] libevdev_jll v1.13.4+0
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  [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
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  [dfaa095f] x265_jll v4.1.0+0
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  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
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  [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
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
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  [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
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+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 ⌃ 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.