E5 Work-Precision Diagrams
using OrdinaryDiffEq, DiffEqDevTools, Sundials, Plots, ODEInterfaceDiffEq, LSODA
using LinearAlgebra, StaticArrays, RecursiveFactorization
using OrdinaryDiffEqFIRK
gr()
# E5 Problem from Stiff Test Set
# Exact implementation based on the Fortran feval subroutine
# 4-dimensional stiff ODE system with widely separated rate constants
function e5!(du, u, p, t)
# E5 problem - exact translation from Fortran feval subroutine:
# prod1=7.89D-10*y(1)
# prod2=1.1D7*y(1)*y(3)
# prod3=1.13D9*y(2)*y(3)
# prod4=1.13D3*y(4)
# f(1)=-prod1-prod2
# f(2)=prod1-prod3
# f(4)=prod2-prod4
# f(3)=f(2)-f(4)
prod1 = 7.89e-10 * u[1]
prod2 = 1.1e7 * u[1] * u[3]
prod3 = 1.13e9 * u[2] * u[3]
prod4 = 1.13e3 * u[4]
du[1] = -prod1 - prod2 # f(1)
du[2] = prod1 - prod3 # f(2)
du[4] = prod2 - prod4 # f(4)
du[3] = du[2] - du[4] # f(3) = f(2) - f(4)
end
function e5(u, p, t)
# E5 problem - exact translation from Fortran feval subroutine:
# prod1=7.89D-10*y(1)
# prod2=1.1D7*y(1)*y(3)
# prod3=1.13D9*y(2)*y(3)
# prod4=1.13D3*y(4)
# f(1)=-prod1-prod2
# f(2)=prod1-prod3
# f(4)=prod2-prod4
# f(3)=f(2)-f(4)
prod1 = 7.89e-10 * u[1]
prod2 = 1.1e7 * u[1] * u[3]
prod3 = 1.13e9 * u[2] * u[3]
prod4 = 1.13e3 * u[4]
du1 = -prod1 - prod2 # f(1)
du2 = prod1 - prod3 # f(2)
du4 = prod2 - prod4 # f(4)
du3 = du2 - du4 # f(3) = f(2) - f(4)
SA[du1,du2,du3,du4]
end
# Initial conditions - system starts with all mass in species A
u0 = [1.76e-3, 0.0, 0.0, 0.0]
# Time span - long enough to see both fast and slow time scales
tspan = (0.0, 1e10)
prob = ODEProblem{true, SciMLBase.FullSpecialize}(e5!, u0, tspan)
probstatic = ODEProblem{false}(e5, SVector{4}(u0), tspan)
probbig = ODEProblem{true, SciMLBase.FullSpecialize}(e5!, big.(u0), big.(tspan))
# Generate reference solution
sol = solve(probbig, Rodas5P(), abstol=1/10^60, reltol=1/10^30)
probs = [prob, probstatic]
test_sol = [sol, sol]
abstols = [1.7e-28 for i in 1:2]
reltols = 10.0 .^ -(6 .+ (1:2)./4)Error: ArgumentError: Package OrdinaryDiffEqFIRK not found in current path.
- Run `import Pkg; Pkg.add("OrdinaryDiffEqFIRK")` to install the OrdinaryDi
ffEqFIRK package.ylabels = 10.0 .^ (-30:2:0)
plot(sol, xscale=:log10, yscale=:log10, tspan = (1e-5, 1e10), labels=["A" "B" "C" "D"], yticks = ylabels)Error: UndefVarError: `sol` not definedBenchmarks
setups = [Dict(:alg=>Rosenbrock23()),
Dict(:alg=>Rosenbrock23(), :prob_choice => 2),
Dict(:alg=>lsoda()),
Dict(:alg=>Rodas5P()),
Dict(:alg=>Rodas5P(), :prob_choice => 2),
Dict(:alg=>FBDF()),
Dict(:alg=>QNDF()),
Dict(:alg=>TRBDF2()),
Dict(:alg=>CVODE_BDF()),
#Dict(:alg=>rodas()),
#Dict(:alg=>radau()),
Dict(:alg=>RadauIIA5()),
Dict(:alg=>ROS34PW1a()),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups; error_estimate = :l2, verbose=false, dense=false,
save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)Error: UndefVarError: `test_sol` not definedsetups = [Dict(:alg=>FBDF()),
Dict(:alg=>QNDF()),
Dict(:alg=>CVODE_BDF()),
Dict(:alg=>lsoda()),
Dict(:alg=>ddebdf()),
Dict(:alg=>Rodas5P()),
Dict(:alg=>Rodas5P(), :prob_choice => 2),
Dict(:alg=>rodas()),
Dict(:alg=>radau()),
Dict(:alg=>RadauIIA5()),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups; verbose=false, dense=false,
save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)Error: UndefVarError: `test_sol` not definedsetups = [Dict(:alg=>Kvaerno4()),
Dict(:alg=>Kvaerno5()),
Dict(:alg=>CVODE_BDF()),
Dict(:alg=>KenCarp4()),
Dict(:alg=>KenCarp47()),
Dict(:alg=>KenCarp47(), :prob_choice => 2),
Dict(:alg=>KenCarp5()),
Dict(:alg=>lsoda()),
Dict(:alg=>Rodas5P()),
Dict(:alg=>Rodas5P(), :prob_choice => 2),
Dict(:alg=>ImplicitEulerExtrapolation(min_order = 4, init_order = 11,threading = OrdinaryDiffEq.PolyesterThreads())),
Dict(:alg=>ImplicitEulerExtrapolation(min_order = 4, init_order = 11,threading = false)),
Dict(:alg=>ImplicitEulerBarycentricExtrapolation(min_order = 4, init_order = 11, threading = OrdinaryDiffEq.PolyesterThreads())),
Dict(:alg=>ImplicitEulerBarycentricExtrapolation(min_order = 4, init_order = 11, threading = false)),
Dict(:alg=>ImplicitHairerWannerExtrapolation(min_order = 3, init_order = 111,threading = OrdinaryDiffEq.PolyesterThreads())),
Dict(:alg=>ImplicitHairerWannerExtrapolation(min_order = 3, init_order = 11,threading = false)),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups; verbose=false, dense=false,
save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)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/StiffODE","E5.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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/StiffODE/Project.toml`
[2169fc97] AlgebraicMultigrid v1.1.0
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[5a33fad7] GeometricIntegratorsDiffEq v0.2.5
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[7ed4a6bd] LinearSolve v3.18.2
⌃ [94925ecb] MethodOfLines v0.11.8
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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:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/StiffODE/Manifest.toml`
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[892a3eda] StringManipulation v0.4.1
[09ab397b] StructArrays v0.7.1
[c3572dad] Sundials v4.28.0
[2efcf032] SymbolicIndexingInterface v0.3.41
[19f23fe9] SymbolicLimits v0.2.2
[d1185830] SymbolicUtils v3.29.0
[0c5d862f] Symbolics v6.43.0
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.12.1
[ed4db957] TaskLocalValues v0.1.2
[62fd8b95] TensorCore v0.1.1
[8ea1fca8] TermInterface v2.0.0
[1c621080] TestItems v1.0.0
[8290d209] ThreadingUtilities v0.5.5
[a759f4b9] TimerOutputs v0.5.29
[c751599d] ToeplitzMatrices v0.8.5
[3bb67fe8] TranscodingStreams v0.11.3
[d5829a12] TriangularSolve v0.2.1
[410a4b4d] Tricks v0.1.10
[781d530d] TruncatedStacktraces v1.4.0
⌃ [5c2747f8] URIs v1.6.0
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
[1986cc42] Unitful v1.23.1
[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
[897b6980] WeakValueDicts v0.1.0
[44d3d7a6] Weave v0.10.12
[efce3f68] WoodburyMatrices v1.0.0
[ddb6d928] YAML v0.4.14
[c2297ded] ZMQ v1.4.1
[6e34b625] Bzip2_jll v1.0.9+0
[83423d85] Cairo_jll v1.18.5+0
[ee1fde0b] Dbus_jll v1.16.2+0
[47200ebd] DecFP_jll v2.0.3+1
[2702e6a9] EpollShim_jll v0.0.20230411+1
[2e619515] Expat_jll v2.6.5+0
⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
[f5851436] FFTW_jll v3.3.11+0
[34b6f7d7] FastTransforms_jll v0.6.3+0
[a3f928ae] Fontconfig_jll v2.16.0+0
[d7e528f0] FreeType2_jll v2.13.4+0
[559328eb] FriBidi_jll v1.0.17+0
[0656b61e] GLFW_jll v3.4.0+2
[d2c73de3] GR_jll v0.73.17+0
[b0724c58] GettextRuntime_jll v0.22.4+0
[f8c6e375] Git_jll v2.50.0+0
[7746bdde] Glib_jll v2.84.3+0
[3b182d85] Graphite2_jll v1.3.15+0
[0234f1f7] HDF5_jll v1.14.6+0
[2e76f6c2] HarfBuzz_jll v8.5.1+0
[e33a78d0] Hwloc_jll v2.12.1+0
[1d5cc7b8] IntelOpenMP_jll v2025.0.4+0
[aacddb02] JpegTurbo_jll v3.1.1+0
⌃ [c1c5ebd0] LAME_jll v3.100.2+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 v2.41.0+0
[89763e89] Libtiff_jll v4.7.1+0
[38a345b3] Libuuid_jll v2.41.0+0
[856f044c] MKL_jll v2025.0.1+1
[7cb0a576] MPICH_jll v4.3.1+0
[f1f71cc9] MPItrampoline_jll v5.5.4+0
[9237b28f] MicrosoftMPI_jll v10.1.4+3
[c771fb93] ODEInterface_jll v0.0.1+0
⌃ [e7412a2a] Ogg_jll v1.3.5+1
[fe0851c0] OpenMPI_jll v5.0.8+0
[9bd350c2] OpenSSH_jll v10.0.1+0
⌃ [458c3c95] OpenSSL_jll v3.5.0+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
⌃ [91d4177d] Opus_jll v1.3.3+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 v1.23.1+1
⌅ [02c8fc9c] XML2_jll v2.13.6+1
[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
[477f73a3] libaec_jll v1.1.3+0
[a4ae2306] libaom_jll v3.11.0+0
⌅ [0ac62f75] libass_jll v0.15.2+0
[1183f4f0] libdecor_jll v0.2.2+0
[2db6ffa8] libevdev_jll v1.13.4+0
⌃ [f638f0a6] libfdk_aac_jll v2.0.3+0
[36db933b] libinput_jll v1.28.1+0
⌃ [b53b4c65] libpng_jll v1.6.49+0
[a9144af2] libsodium_jll v1.0.21+0
⌃ [f27f6e37] libvorbis_jll v1.3.7+2
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.0.0+0
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
[d8fb68d0] xkbcommon_jll v1.9.2+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 v0.6.4
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.10.0
[de0858da] Printf
[9abbd945] Profile
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays v1.10.0
[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
[781609d7] GMP_jll v6.2.1+6
[deac9b47] LibCURL_jll v8.4.0+0
[e37daf67] LibGit2_jll v1.6.4+0
[29816b5a] LibSSH2_jll v1.11.0+1
[3a97d323] MPFR_jll v4.2.0+1
[c8ffd9c3] MbedTLS_jll v2.28.2+1
[14a3606d] MozillaCACerts_jll v2023.1.10
[4536629a] OpenBLAS_jll v0.3.23+4
[05823500] OpenLibm_jll v0.8.1+4
[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`