Three Body Work-Precision Diagrams

using OrdinaryDiffEq, ODE, ODEInterfaceDiffEq, LSODA, Sundials, DiffEqDevTools, StaticArrays
using Plots; gr()

## Define the ThreeBody Problem
const threebody_μ = parse(Float64,"0.012277471")
const threebody_μ′ = 1 - threebody_μ

function f(du,u,p,t)
  @inbounds begin
    # 1 = y₁
    # 2 = y₂
    # 3 = y₁'
    # 4 = y₂'
    D₁ = ((u[1]+threebody_μ)^2 + u[2]^2)^(3/2)
    D₂ = ((u[1]-threebody_μ′)^2 + u[2]^2)^(3/2)
    du[1] = u[3]
    du[2] = u[4]
    du[3] = u[1] + 2u[4] - threebody_μ′*(u[1]+threebody_μ)/D₁ - threebody_μ*(u[1]-threebody_μ′)/D₂
    du[4] = u[2] - 2u[3] - threebody_μ′*u[2]/D₁ - threebody_μ*u[2]/D₂

function f(u,p,t)
  @inbounds begin
    # 1 = y₁
    # 2 = y₂
    # 3 = y₁'
    # 4 = y₂'
    D₁ = ((u[1]+threebody_μ)^2 + u[2]^2)^(3/2)
    D₂ = ((u[1]-threebody_μ′)^2 + u[2]^2)^(3/2)
    du1 = u[3]
    du2 = u[4]
    du3 = u[1] + 2u[4] - threebody_μ′*(u[1]+threebody_μ)/D₁ - threebody_μ*(u[1]-threebody_μ′)/D₂
    du4 = u[2] - 2u[3] - threebody_μ′*u[2]/D₁ - threebody_μ*u[2]/D₂

t₀ = 0.0; T = parse(Float64,"17.0652165601579625588917206249")
tspan = (t₀,2T)

prob = ODEProblem{true, SciMLBase.FullSpecialize}(f,[0.994, 0.0, 0.0, parse(Float64,"-2.00158510637908252240537862224")],tspan)
probstatic = ODEProblem{false}(f,SA[0.994, 0.0, 0.0, parse(Float64,"-2.00158510637908252240537862224")],tspan)

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];

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

See that it's periodic in the chosen timespan:

sol = solve(prob,Vern9(),abstol=1e-14,reltol=1e-14)
@show sol[1] - sol[end]
@show sol[end] - prob.u0;
sol[1] - sol[end] = [1.2515211089692002e-11, 3.706335003529717e-11, 6.04785
4328438363e-9, 1.9479733381899678e-9]
sol[end] - prob.u0 = [-1.2515211089692002e-11, -3.706335003529717e-11, -6.0
47854328438363e-9, -1.9479733381899678e-9]

This three-body problem is known to be a tough problem. Let's see how the algorithms do at standard tolerances.

5th Order Runge-Kutta Methods

setups = [Dict(:alg=>DP5())
          #Dict(:alg=>ode45()) #fails
          Dict(:alg=>Tsit5(), :prob_choice => 2)
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,save_everystep=false,numruns=100)

Full save, but no dense

setups = [Dict(:alg=>DP5(),:dense=>false)
          #Dict(:alg=>ode45()) # Fails
          Dict(:alg=>Tsit5(),:dense=>false, :prob_choice => 2)
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,numruns=100)


setups = [Dict(:alg=>DP5())
          #Dict(:alg=>ode45()) #fails
          Dict(:alg=>Tsit5(), :prob_choice => 2)
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,numruns=100)

In these tests we see that most of the algorithms are close,with BS5 and DP5 showing much better than Tsit5. ode45 errors.

Higher Order Algorithms

setups = [Dict(:alg=>DP5())
          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)

wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,dense=false,numruns=100,verbose=false)

wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,numruns=100)

In this test we see Vern7 and Vern8 shine.

Other Algorithms

Once again we separate ODE.jl because it fails. We also separate Sundials' CVODE_Adams since it fails at high tolerances.

#setups = [Dict(:alg=>ode78())
#          Dict(:alg=>VCABM())
#          Dict(:alg=>CVODE_Adams())];
#wp = WorkPrecisionSet(prob,abstols,reltols,setups;appxsol=test_sol,dense=false,numruns=100)
setups = [Dict(:alg=>DP5())
          Dict(:alg=>Vern8(), :prob_choice => 2)
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,save_everystep=false,numruns=100)

Again, on cheap function calculations the Adams methods are shown to not be efficient once the error is sufficiently small. Also, as seen in other places, the extrapolation methods do not fare as well as the Runge-Kutta methods.

Comparison with Non-RK methods

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

abstols = 1.0 ./ 10.0 .^ (3:13); reltols = 1.0 ./ 10.0 .^ (0:10);
setups = [Dict(:alg=>Tsit5())
          Dict(:alg=>Vern9(), :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","Vern9 Static","AitkenNeville","Midpoint Deuflhard","Midpoint Hairer Wanner"]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;appxsol=test_sol,names=solnames,

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,

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,


As in the other tests, the OrdinaryDiffEq.jl algorithms with the Verner Efficient methods are the most efficient solvers at stringent tolerances for most of the tests, while the order 5 methods do well at cruder tolerances. ODE.jl fails to run the test problems without erroring.


These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization

To locally run this benchmark, do the following commands:

using SciMLBenchmarks

Computer Information:

Julia Version 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
  BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
  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 v2.32.0
  [7f56f5a3] LSODA v0.7.0
  [c030b06c] ODE v2.15.0
  [54ca160b] ODEInterface v0.5.0
  [09606e27] ODEInterfaceDiffEq v3.11.0
  [1dea7af3] OrdinaryDiffEq v6.27.0
  [65888b18] ParameterizedFunctions v5.14.0
  [91a5bcdd] Plots v1.33.0
  [31c91b34] SciMLBenchmarks v0.1.1
  [90137ffa] StaticArrays v1.5.7
  [c3572dad] Sundials v4.10.1
  [9a3f8284] Random

And the full manifest:

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  [e7412a2a] Ogg_jll v1.3.5+1
  [458c3c95] OpenSSL_jll v1.1.17+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [91d4177d] Opus_jll v1.3.2+0
  [2f80f16e] PCRE_jll v8.44.0+0
  [30392449] Pixman_jll v0.40.1+0
  [ea2cea3b] Qt5Base_jll v5.15.3+1
  [f50d1b31] Rmath_jll v0.3.0+0
  [fb77eaff] Sundials_jll v5.2.1+0
  [a2964d1f] Wayland_jll v1.19.0+0
  [2381bf8a] Wayland_protocols_jll v1.25.0+0
  [02c8fc9c] XML2_jll v2.9.14+0
  [aed1982a] XSLT_jll v1.1.34+0
  [4f6342f7] Xorg_libX11_jll v1.6.9+4
  [0c0b7dd1] Xorg_libXau_jll v1.0.9+4
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.3+4
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [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 v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.0+3
  [c7cfdc94] Xorg_libxcb_jll v1.13.0+3
  [cc61e674] Xorg_libxkbfile_jll v1.1.0+4
  [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 v1.4.2+4
  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
  [c5fb5394] Xorg_xtrans_jll v1.4.0+3
  [8f1865be] ZeroMQ_jll v4.3.4+0
  [3161d3a3] Zstd_jll v1.5.2+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [b53b4c65] libpng_jll v1.6.38+0
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+0
  [0dad84c5] ArgTools
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [b27032c2] LibCURL
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions
  [44cfe95a] Pkg
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML
  [a4e569a6] Tar
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll
  [deac9b47] LibCURL_jll
  [29816b5a] LibSSH2_jll
  [c8ffd9c3] MbedTLS_jll
  [14a3606d] MozillaCACerts_jll
  [4536629a] OpenBLAS_jll
  [05823500] OpenLibm_jll
  [efcefdf7] PCRE2_jll
  [bea87d4a] SuiteSparse_jll
  [83775a58] Zlib_jll
  [8e850b90] libblastrampoline_jll
  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll