Three Body Work-Precision Diagrams

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

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

f = (du,u,p,t) -> begin
  @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
  end
end

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

prob = ODEProblem(f,[0.994, 0.0, 0.0, parse(Float64,"-2.00158510637908252240537862224")],tspan)

test_sol = TestSolution(T,[prob.u0])
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]
apr = appxtrue(sol,test_sol)
@show sol[end]
@show apr.u[end]
@show apr.errors
sol[end] = [0.9939999999874848, -3.706335003529717e-11, -6.047854328438363e
-9, -2.0015851083270557]
apr.u[end] = [0.9939999999874848, -3.706335003529717e-11, -6.04785432843836
3e-9, -2.0015851083270557]
apr.errors = Dict(:final => 2.01135155693833e-9)
Dict{Symbol, Float64} with 1 entry:
  :final => 2.01135e-9

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

5th Order Runge-Kutta Methods

setups = [Dict(:alg=>DP5())
          #Dict(:alg=>ode45()) #fails
          Dict(:alg=>BS5())
          Dict(:alg=>Tsit5())
          Dict(:alg=>dopri5())];
wp = WorkPrecisionSet(prob,abstols,reltols,setups;appxsol=test_sol,save_everystep=false,numruns=100)
plot(wp)

Full save, but no dense

setups = [Dict(:alg=>DP5(),:dense=>false)
          #Dict(:alg=>ode45()) # Fails
          Dict(:alg=>BS5(),:dense=>false)
          Dict(:alg=>Tsit5(),:dense=>false)
          Dict(:alg=>dopri5())];
wp = WorkPrecisionSet(prob,abstols,reltols,setups;appxsol=test_sol,numruns=100)
plot(wp)

Dense

setups = [Dict(:alg=>DP5())
          #Dict(:alg=>ode45()) #fails
          Dict(:alg=>BS5())
          Dict(:alg=>Tsit5())
          Dict(:alg=>dopri5())];
wp = WorkPrecisionSet(prob,abstols,reltols,setups;appxsol=test_sol,numruns=100)
plot(wp)

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())
          Dict(:alg=>Vern7())
          Dict(:alg=>TanYam7())
          Dict(:alg=>Vern8())
          Dict(:alg=>DP8())
          Dict(:alg=>dop853())
          Dict(:alg=>Vern9())];
wp = WorkPrecisionSet(prob,abstols,reltols,setups;appxsol=test_sol,save_everystep=false,numruns=100)
plot(wp)

setups = [Dict(:alg=>DP5())
          Dict(:alg=>Vern6())
          Dict(:alg=>Vern7())
          Dict(:alg=>TanYam7())
          Dict(:alg=>Vern8())
          Dict(:alg=>DP8())
          Dict(:alg=>dop853())
          Dict(:alg=>Vern9())];
wp = WorkPrecisionSet(prob,abstols,reltols,setups;appxsol=test_sol,dense=false,numruns=100,verbose=false)
plot(wp)

setups = [Dict(:alg=>DP5())
          Dict(:alg=>Vern6())
          Dict(:alg=>Vern7())
          Dict(:alg=>TanYam7())
          Dict(:alg=>Vern8())
          Dict(:alg=>DP8())
          Dict(:alg=>dop853())
          Dict(:alg=>Vern9())];
wp = WorkPrecisionSet(prob,abstols,reltols,setups;appxsol=test_sol,numruns=100)
plot(wp)

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=>lsoda())
          Dict(:alg=>Vern8())
          Dict(:alg=>ddeabm())
          Dict(:alg=>odex())
          Dict(:alg=>ARKODE(Sundials.Explicit(),order=6))
    ];
wp = WorkPrecisionSet(prob,abstols,reltols,setups;appxsol=test_sol,save_everystep=false,numruns=100)
plot(wp)

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

Conclusion

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.

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","ThreeBody_wpd.jmd")

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
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
Environment:
  JULIA_CPU_THREADS = 128
  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 v2.30.0
  [7f56f5a3] LSODA v0.7.0
  [c030b06c] ODE v2.13.0
  [54ca160b] ODEInterface v0.5.0
  [09606e27] ODEInterfaceDiffEq v3.10.1
  [1dea7af3] OrdinaryDiffEq v6.19.0
  [65888b18] ParameterizedFunctions v5.13.2
  [91a5bcdd] Plots v1.31.4
  [31c91b34] SciMLBenchmarks v0.1.0
  [c3572dad] Sundials v4.9.4
  [9a3f8284] Random

And the full manifest:

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