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
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())]
solnames = ["OrdinaryDiffEq";"ODE";"ODEInterface";"Sundials ARKODE";"OrdinaryDiffEq Tsit5"]
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)

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 1.10.4
Commit 48d4fd48430 (2024-06-04 10:41 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/NonStiffODE/Project.toml`
  [f3b72e0c] DiffEqDevTools v2.44.4
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  [91a5bcdd] Plots v1.40.5
  [31c91b34] SciMLBenchmarks v0.1.3
  [90137ffa] StaticArrays v1.9.7
  [c3572dad] Sundials v4.24.0
  [9a3f8284] Random

And the full manifest:

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  [5c2747f8] URIs v1.5.1
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.21.0
  [45397f5d] UnitfulLatexify v1.6.4
  [a7c27f48] Unityper v0.1.6
  [41fe7b60] Unzip v0.2.0
  [3d5dd08c] VectorizationBase v0.21.70
  [81def892] VersionParsing v1.3.0
  [19fa3120] VertexSafeGraphs v0.2.0
  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.12
  [c2297ded] ZMQ v1.2.6
  [6e34b625] Bzip2_jll v1.0.8+1
  [83423d85] Cairo_jll v1.18.0+2
  [2702e6a9] EpollShim_jll v0.0.20230411+0
  [2e619515] Expat_jll v2.6.2+0
⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
  [a3f928ae] Fontconfig_jll v2.13.96+0
  [d7e528f0] FreeType2_jll v2.13.2+0
  [559328eb] FriBidi_jll v1.0.14+0
  [0656b61e] GLFW_jll v3.4.0+0
  [d2c73de3] GR_jll v0.73.7+0
  [78b55507] Gettext_jll v0.21.0+0
  [f8c6e375] Git_jll v2.44.0+2
  [7746bdde] Glib_jll v2.80.2+0
  [3b182d85] Graphite2_jll v1.3.14+0
  [2e76f6c2] HarfBuzz_jll v2.8.1+1
  [1d5cc7b8] IntelOpenMP_jll v2024.2.0+0
  [aacddb02] JpegTurbo_jll v3.0.3+0
  [c1c5ebd0] LAME_jll v3.100.2+0
⌅ [88015f11] LERC_jll v3.0.0+1
  [1d63c593] LLVMOpenMP_jll v15.0.7+0
  [aae0fff6] LSODA_jll v0.1.2+0
  [dd4b983a] LZO_jll v2.10.2+0
⌅ [e9f186c6] Libffi_jll v3.2.2+1
  [d4300ac3] Libgcrypt_jll v1.8.11+0
  [7e76a0d4] Libglvnd_jll v1.6.0+0
  [7add5ba3] Libgpg_error_jll v1.49.0+0
  [94ce4f54] Libiconv_jll v1.17.0+0
  [4b2f31a3] Libmount_jll v2.40.1+0
⌅ [89763e89] Libtiff_jll v4.5.1+1
  [38a345b3] Libuuid_jll v2.40.1+0
  [856f044c] MKL_jll v2024.2.0+0
  [c771fb93] ODEInterface_jll v0.0.1+0
  [e7412a2a] Ogg_jll v1.3.5+1
  [458c3c95] OpenSSL_jll v3.0.14+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [91d4177d] Opus_jll v1.3.2+0
  [30392449] Pixman_jll v0.43.4+0
  [c0090381] Qt6Base_jll v6.7.1+1
  [629bc702] Qt6Declarative_jll v6.7.1+2
  [ce943373] Qt6ShaderTools_jll v6.7.1+1
  [e99dba38] Qt6Wayland_jll v6.7.1+1
  [f50d1b31] Rmath_jll v0.4.2+0
⌅ [fb77eaff] Sundials_jll v5.2.2+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.21.0+1
  [2381bf8a] Wayland_protocols_jll v1.31.0+0
  [02c8fc9c] XML2_jll v2.13.1+0
  [aed1982a] XSLT_jll v1.1.41+0
  [ffd25f8a] XZ_jll v5.4.6+0
  [f67eecfb] Xorg_libICE_jll v1.1.1+0
  [c834827a] Xorg_libSM_jll v1.2.4+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 v1.3.6+0
  [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 v0.9.11+0
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.1+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.0+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.2+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+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 v4.3.5+0
  [3161d3a3] Zstd_jll v1.5.6+0
  [35ca27e7] eudev_jll v3.2.9+0
⌅ [214eeab7] fzf_jll v0.43.0+0
  [1a1c6b14] gperf_jll v3.1.1+0
  [a4ae2306] libaom_jll v3.9.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.43+1
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.7+2
  [009596ad] mtdev_jll v1.1.6+0
  [1317d2d5] oneTBB_jll v2021.12.0+0
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+1
  [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
  [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
  [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 v0.8.1+2
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.8.0+1
  [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`