100 Independent Linear Work-Precision Diagrams

For these tests we will solve a diagonal 100 independent linear differential equations. This will demonstrate the efficiency of the implementation of the methods for handling large systems, since the system is both large enough that array handling matters, but f is cheap enough that it is not simply a game of calculating f as few times as possible. We will be mostly looking at the efficiency of the work-horse Dormand-Prince Order 4/5 Pairs: one from DifferentialEquations.jl (DP5), one from ODE.jl rk45, one from ODEInterface (Hairer's famous dopri5, and one from SUNDIALS' ARKODE suite.

Also included is Tsit5. While all other ODE programs have gone with the traditional choice of using the Dormand-Prince 4/5 pair as the default, DifferentialEquations.jl uses Tsit5 as one of the default algorithms. It's a very new (2011) and not widely known, but the theory and the implementation shows it's more efficient than DP5. Thus we include it just to show off how re-designing a library from the ground up in a language for rapid code and rapid development has its advantages.

Setup

using OrdinaryDiffEq, Sundials, DiffEqDevTools, Plots, ODEInterfaceDiffEq, LSODA, OrdinaryDiffEqSIMDRK
using Random
Random.seed!(123)
gr()
# 2D Linear ODE
function f(du, u, p, t)
    @inbounds for i in eachindex(u)
        du[i] = 1.01*u[i]
    end
end
function f_analytic(u₀, p, t)
    u₀*exp(1.01*t)
end
tspan = (0.0, 10.0)
prob = ODEProblem(ODEFunction{true, SciMLBase.FullSpecialize}(f, analytic = f_analytic), rand(100, 100), tspan)

abstols = 1.0 ./ 10.0 .^ (3:13)
reltols = 1.0 ./ 10.0 .^ (0:10);

Speed Baseline

First a baseline. These are all testing the same Dormand-Prince order 5/4 algorithm of each package. While all the same Runge-Kutta tableau, they exhibit different behavior due to different choices of adaptive timestepping algorithms and tuning. First we will test with all extra saving features are turned off to put DifferentialEquations.jl in "speed mode".

setups = [Dict(:alg=>DP5())
          Dict(:alg=>ode45())
          Dict(:alg=>dopri5())
          Dict(:alg=>ARKODE(Sundials.Explicit(), etable = Sundials.DORMAND_PRINCE_7_4_5))
          Dict(:alg=>Tsit5())
          Dict(:alg=>MER5v2())
          Dict(:alg=>MER6v2())
          Dict(:alg=>RK6v4())]
solnames = ["OrdinaryDiffEq"; "ODE"; "ODEInterface"; "Sundials ARKODE";
            "OrdinaryDiffEq Tsit5"; "MER5v2"; "MER6v2"; "RK6v4"]
wp = WorkPrecisionSet(
    prob, abstols, reltols, setups; names = solnames, save_everystep = false, numruns = 100)
plot(wp)

Error: UndefVarError: `ode45` not defined

Full Saving

setups = [Dict(:alg=>DP5(), :dense=>false)
          Dict(:alg=>ode45(), :dense=>false)
          Dict(:alg=>dopri5()) # dense=false by default: no nonlinear interpolation
          Dict(:alg=>ARKODE(Sundials.Explicit(), etable = Sundials.DORMAND_PRINCE_7_4_5), :dense=>false)
          Dict(:alg=>Tsit5(), :dense=>false)]
solnames = ["OrdinaryDiffEq"; "ODE"; "ODEInterface"; "Sundials ARKODE";
            "OrdinaryDiffEq Tsit5"]
wp = WorkPrecisionSet(prob, abstols, reltols, setups; names = solnames, numruns = 100)
plot(wp)

Error: UndefVarError: `ode45` not defined

Continuous Output

Now we include continuous output. This has a large overhead because at every timepoint the matrix of rates k has to be deep copied.

setups = [Dict(:alg=>DP5())
          Dict(:alg=>ode45())
          Dict(:alg=>dopri5())
          Dict(:alg=>ARKODE(Sundials.Explicit(), etable = Sundials.DORMAND_PRINCE_7_4_5))
          Dict(:alg=>Tsit5())]
solnames = ["OrdinaryDiffEq"; "ODE"; "ODEInterface"; "Sundials ARKODE";
            "OrdinaryDiffEq Tsit5"]
wp = WorkPrecisionSet(prob, abstols, reltols, setups; names = solnames, numruns = 100)
plot(wp)

Error: UndefVarError: `ode45` not defined

Other Runge-Kutta Algorithms

Now let's test it against a smattering of other Runge-Kutta algorithms. First we will test it with all overheads off. Let's do the Order 5 (and the 2/3 pair) algorithms:

setups = [Dict(:alg=>DP5())
          Dict(:alg=>BS3())
          Dict(:alg=>BS5())
          Dict(:alg=>Tsit5())]
wp = WorkPrecisionSet(prob, abstols, reltols, setups; save_everystep = false, numruns = 100)
plot(wp)

Higher Order

Now let's see how OrdinaryDiffEq.jl fairs with some higher order algorithms:

setups = [Dict(:alg=>DP5())
          Dict(:alg=>Vern6())
          Dict(:alg=>TanYam7())
          Dict(:alg=>Vern7())
          Dict(:alg=>Vern8())
          Dict(:alg=>DP8())
          Dict(:alg=>Vern9())]
wp = WorkPrecisionSet(prob, abstols, reltols, setups; save_everystep = false, numruns = 100)
plot(wp)

Higher Order With Many Packages

Now we test OrdinaryDiffEq against the high order methods of the other packages:

setups = [Dict(:alg=>DP5())
          Dict(:alg=>Vern7())
          Dict(:alg=>dop853())
          Dict(:alg=>ode78())
          Dict(:alg=>odex())
          Dict(:alg=>lsoda())
          Dict(:alg=>ddeabm())
          Dict(:alg=>ARKODE(Sundials.Explicit(), order = 8))
          Dict(:alg=>CVODE_Adams())]
wp = WorkPrecisionSet(prob, abstols, reltols, setups; save_everystep = false, numruns = 100)
plot(wp)

Error: UndefVarError: `ode78` not defined

Interpolation Error

Now we will look at the error using an interpolation measurement instead of at the timestepping points. Since the DifferentialEquations.jl algorithms have higher order interpolants than the ODE.jl algorithms, one would expect this would magnify the difference. First the order 4/5 comparison:

setups = [Dict(:alg=>DP5())
          #Dict(:alg=>ode45())
          Dict(:alg=>Tsit5())]
wp = WorkPrecisionSet(prob, abstols, reltols, setups; error_estimate = :L2,
    dense_errors = true, numruns = 100)
plot(wp)

Note that all of ODE.jl uses a 3rd order Hermite interpolation, while the DifferentialEquations algorithms interpolations which are specialized to the algorithm. For example, DP5 and Tsit5 both use "free" order 4 interpolations, which are both as fast as the Hermite interpolation while achieving far less error. At higher order:

setups = [Dict(:alg=>DP5())
          Dict(:alg=>Vern7())
          #Dict(:alg=>ode78())
          ]
wp = WorkPrecisionSet(prob, abstols, reltols, setups; error_estimate = :L2,
    dense_errors = true, numruns = 100)
plot(wp)

Comparison with Fixed Timestep RK4

Let's run the first benchmark but add some fixed timestep RK4 methods to see the difference:

abstols = 1.0 ./ 10.0 .^ (3:13)
reltols = 1.0 ./ 10.0 .^ (0:10);
dts = [1, 1/2, 1/4, 1/10, 1/20, 1/40, 1/60, 1/80, 1/100, 1/140, 1/240]
setups = [Dict(:alg=>DP5())
          Dict(:alg=>ode45())
          Dict(:alg=>dopri5())
          Dict(:alg=>RK4(), :dts=>dts)
          Dict(:alg=>Tsit5())]
solnames = ["DifferentialEquations"; "ODE"; "ODEInterface"; "DifferentialEquations RK4";
            "DifferentialEquations Tsit5"]
wp = WorkPrecisionSet(prob, abstols, reltols, setups; names = solnames,
    save_everystep = false, verbose = false, numruns = 100)
plot(wp)

Error: UndefVarError: `ode45` not defined

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=>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", "AitkenNeville",
    "Midpoint Deuflhard", "Midpoint Hairer Wanner"]
wp = WorkPrecisionSet(prob, abstols, reltols, setups; names = solnames,
    save_everystep = false, verbose = false, 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(prob, abstols, reltols, setups; names = solnames,
    save_everystep = false, verbose = false, 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(prob, abstols, reltols, setups; names = solnames,
    save_everystep = false, verbose = false, numruns = 100)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (12:15)
reltols = 1.0 ./ 10.0 .^ (9:12)

setups = [Dict(:alg=>Tsit5())
          Dict(:alg=>Vern9())
          Dict(:alg=>VCABM())
          #Dict(:alg=>AitkenNeville(threading = OrdinaryDiffEq.PolyesterThreads()))
          Dict(:alg=>ExtrapolationMidpointDeuflhard(threading = OrdinaryDiffEq.PolyesterThreads()))
          Dict(:alg=>ExtrapolationMidpointHairerWanner(threading = OrdinaryDiffEq.PolyesterThreads()))
          Dict(:alg=>odex())
          Dict(:alg=>dop853())
          Dict(:alg=>CVODE_Adams())]

wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    save_everystep = false, verbose = false, numruns = 100)
plot(wp)

Error: UndefVarError: `PolyesterThreads` not defined

Conclusion

DifferentialEquations's default choice of Tsit5 does well for quick and easy solving at normal tolerances. However, at low tolerances the higher order algorithms are faster. In every case, the DifferentialEquations algorithms are far in the lead, many times an order of magnitude faster than the competitors. Vern7 with its included 7th order interpolation looks to be a good workhorse for scientific computing in floating point range. These along with many other benchmarks are why these algorithms were chosen as part of the defaults.

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","linear_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 v6.188.0
  [f3b72e0c] DiffEqDevTools v2.48.0
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  [c3572dad] Sundials v4.28.0
  [0c5d862f] Symbolics v6.54.0
  [37e2e46d] LinearAlgebra
  [9a3f8284] Random
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:

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  [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
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  [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
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  [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
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  [7b1f6079] FileWatching
  [9fa8497b] Future
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  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
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  [de0858da] Printf
  [3fa0cd96] REPL
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  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
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  [14a3606d] MozillaCACerts_jll v2023.1.10
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  [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.