ROBER Work-Precision Diagrams

using OrdinaryDiffEq, DiffEqDevTools, Sundials, ParameterizedFunctions, Plots, ODEInterfaceDiffEq, LSODA
gr()
using LinearAlgebra, StaticArrays, RecursiveFactorization

rober = @ode_def begin
  dy₁ = -k₁*y₁+k₃*y₂*y₃
  dy₂ =  k₁*y₁-k₂*y₂^2-k₃*y₂*y₃
  dy₃ =  k₂*y₂^2
end k₁ k₂ k₃

prob = ODEProblem{true, SciMLBase.FullSpecialize}(rober,[1.0,0.0,0.0],(0.0,1e5),(0.04,3e7,1e4))
probstatic = ODEProblem{false}(rober,SA[1.0,0.0,0.0],(0.0,1e5),(0.04,3e7,1e4))

sol = solve(prob,CVODE_BDF(),abstol=1/10^14,reltol=1/10^14)
sol2 = solve(probstatic,Rodas5P(),abstol=1/10^14,reltol=1/10^14)
probs = [prob,probstatic]
test_sol = [sol,sol2];

abstols = 1.0 ./ 10.0 .^ (4:11)
reltols = 1.0 ./ 10.0 .^ (1:8);

plot(sol,labels=["y1","y2","y3"])

Omissions And Tweaking

The following were omitted from the tests due to convergence failures. ODE.jl's adaptivity is not able to stabilize its algorithms, while GeometricIntegratorsDiffEq has not upgraded to Julia 1.0. GeometricIntegrators.jl's methods used to be either fail to converge at comparable dts (or on some computers errors due to type conversions).

#sol = solve(prob,ode23s()); println("Total ODE.jl steps: $(length(sol))")
#using GeometricIntegratorsDiffEq
#try
#    sol = solve(prob,GIRadIIA3(),dt=1/10)
#catch e
#    println(e)
#end

ARKODE needs a lower nonlinear_convergence_coefficient in order to not diverge.

#sol = solve(prob,ARKODE(nonlinear_convergence_coefficient = 1e-6),abstol=1e-5,reltol=1e-1); # Noisy, output omitted

sol = solve(prob,ARKODE(nonlinear_convergence_coefficient = 1e-7),abstol=1e-5,reltol=1e-1);

Note that 1e-7 matches the value from the Sundials manual which was required for their example to converge on this problem. The default is 1e-1.

#sol = solve(prob,ARKODE(order=3),abstol=1e-4,reltol=1e-1); # Fails to diverge but doesn't finish

#sol = solve(prob,ARKODE(order=5),abstol=1e-4,reltol=1e-1); # Noisy, output omitted

#sol = solve(prob,ARKODE(order=5,nonlinear_convergence_coefficient = 1e-9),abstol=1e-5,reltol=1e-1); # Noisy, output omitted

Additionally, the ROCK methods do not perform well on this benchmark.

setups = [
          #Dict(:alg=>ROCK2())    #Unstable
          #Dict(:alg=>ROCK4())    #needs more iterations
          ]

Any[]

Some of the bad Rosenbrocks fail:

setups = [
  #Dict(:alg=>Hairer4()),
  #Dict(:alg=>Hairer42()),
  #Dict(:alg=>Cash4()),
]

Any[]

The EPIRK and exponential methods also fail:

sol = solve(prob,EXPRB53s3(),dt=2.0^(-8));
sol = solve(prob,EPIRK4s3B(),dt=2.0^(-8));
sol = solve(prob,EPIRK5P2(),dt=2.0^(-8));

Error: InexactError: trunc(Int64, Inf)

PDIRK44 also fails

sol = solve(prob,PDIRK44(),dt=2.0^(-8));

In fact, all non-adaptive methods fail on this problem.

High Tolerances

This is the speed when you just want the answer. ode23s from ODE.jl was removed since it fails. Note that at high tolerances Sundials' CVODE_BDF fails as well so it's excluded from this test.

abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4);
setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Rosenbrock23(), :prob_choice => 2),
          Dict(:alg=>FBDF()),
          Dict(:alg=>QNDF()),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>rodas()),
          Dict(:alg=>lsoda()),
          Dict(:alg=>radau()),
          Dict(:alg=>RadauIIA5()),
          Dict(:alg=>ROS34PW1a()),
          ]
gr()
wp = WorkPrecisionSet(probs,abstols,reltols,setups; verbose=false, dense=false,
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Rosenbrock23(), :prob_choice => 2),
          Dict(:alg=>Kvaerno3()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>KenCarp3()),
          Dict(:alg=>lsoda()),
          # Dict(:alg=>SDIRK2()), # Removed because it's bad
          Dict(:alg=>radau())]
names = ["Rosenbrock23" "Rosenbrock23 Static" "Kvaerno3" "KenCarp4" "TRBDF2" "KenCarp3" "lsoda" "radau"]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;names=names, verbose=false, dense=false,
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Rosenbrock23(), :prob_choice => 2),
          Dict(:alg=>KenCarp5()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>KenCarp4(), :prob_choice => 2),
          Dict(:alg=>KenCarp3()),
          Dict(:alg=>ARKODE(nonlinear_convergence_coefficient = 1e-9,order=5)),
          Dict(:alg=>ARKODE(nonlinear_convergence_coefficient = 1e-8)),
          Dict(:alg=>ARKODE(nonlinear_convergence_coefficient = 1e-7,order=3))
]
names = ["Rosenbrock23" "Rosenbrock23 Static" "KenCarp5" "KenCarp4" "KenCarp4 Static" "KenCarp3" "ARKODE5" "ARKODE4" "ARKODE3"]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
                      names=names, verbose=false, dense=false,
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Rosenbrock23(), :prob_choice => 2),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>ImplicitEulerExtrapolation()),
          Dict(:alg=>ImplicitEulerExtrapolation()),
          Dict(:alg=>ImplicitEulerBarycentricExtrapolation()),
          Dict(:alg=>ImplicitHairerWannerExtrapolation()),
          #Dict(:alg=>ABDF2()), # Maxiters
          Dict(:alg=>FBDF()),
          Dict(:alg=>QNDF()),
          #Dict(:alg=>Exprb43()), #SingularException
          #Dict(:alg=>Exprb32()), #SingularException
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups; verbose=false, dense=false,
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

Timeseries Errors

abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4);
setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Rosenbrock23(), :prob_choice => 2),
          Dict(:alg=>FBDF()),
          Dict(:alg=>QNDF()),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>rodas()),
          Dict(:alg=>lsoda()),
          Dict(:alg=>radau()),
          Dict(:alg=>RadauIIA5()),
          Dict(:alg=>ROS34PW1a()),
          ]
gr()
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)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Rosenbrock23(), :prob_choice => 2),
          Dict(:alg=>Kvaerno3()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>KenCarp3()),
    # Dict(:alg=>SDIRK2()), # Removed because it's bad
          Dict(:alg=>radau())]
names = ["Rosenbrock23" "Rosenbrock23 Static" "Kvaerno3" "KenCarp4" "TRBDF2" "KenCarp3" "radau"]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;names=names, verbose=false, dense=false,
                      appxsol=test_sol,maxiters=Int(1e5),error_estimate=:l2,numruns=10)
plot(wp)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Rosenbrock23(), :prob_choice => 2),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>ImplicitEulerExtrapolation()),
          Dict(:alg=>ImplicitEulerExtrapolation()),
          Dict(:alg=>ImplicitEulerBarycentricExtrapolation()),
          Dict(:alg=>ImplicitHairerWannerExtrapolation()),
          #Dict(:alg=>ABDF2()), # Maxiters
          Dict(:alg=>FBDF()),
          Dict(:alg=>QNDF()),
          #Dict(:alg=>Exprb43()), #SingularException
          #Dict(:alg=>Exprb32()), #SingularException
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;verbose=false,error_estimate=:l2,
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

Low Tolerances

This is the speed at lower tolerances, measuring what's good when accuracy is needed.

abstols = 1.0 ./ 10.0 .^ (7:12)
reltols = 1.0 ./ 10.0 .^ (4:9)

setups = [#Dict(:alg=>Rodas5()),
          Dict(:alg=>FBDF()),
          Dict(:alg=>QNDF()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>ddebdf()),
          Dict(:alg=>Rodas4()),
          Dict(:alg=>Rodas4(), :prob_choice => 2),
          #Dict(:alg=>Rodas5P()),
          Dict(:alg=>rodas()),
          Dict(:alg=>lsoda()),
          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)

setups = [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=>Rodas4()),
          Dict(:alg=>Rodas4(), :prob_choice => 2),
          #Dict(:alg=>Rodas5P()),
          #Dict(:alg=>Rodas5()),
          Dict(:alg=>lsoda()),
          Dict(:alg=>radau()),
          Dict(:alg=>ImplicitEulerExtrapolation()),
          Dict(:alg=>ImplicitEulerBarycentricExtrapolation()),
          Dict(:alg=>ImplicitHairerWannerExtrapolation()),
          ]
wp = WorkPrecisionSet(probs,abstols,reltols,setups; verbose=false, dense=false,
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (10:12)
reltols = 1.0 ./ 10.0 .^ (7:9)

setups = [Dict(:alg=>Rodas4())
          Dict(:alg=>Rodas4(), :prob_choice => 2)
          Dict(:alg=>Rodas5())
          Dict(:alg=>Rodas5(), :prob_choice => 2)
          Dict(:alg=>Rodas5P())
          Dict(:alg=>Rodas5P(), :prob_choice => 2)]
names = ["Rodas4" "Rodas4 Static" "Rodas5" "Rodas5 Static" "Rodas5P" "Rodas5P Static"]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;names=names, verbose=false, dense=false,
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

Multithreading benchmarks with Parallel Extrapolation Methods

#Setting BLAS to one thread to measure gains
LinearAlgebra.BLAS.set_num_threads(1)

abstols = 1.0 ./ 10.0 .^ (10:12)
reltols = 1.0 ./ 10.0 .^ (7:9)

setups = [
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>Rodas4()),
          Dict(:alg=>Rodas4(), :prob_choice => 2),
          Dict(:alg=>Rodas5P()),
          Dict(:alg=>Rodas5P(), :prob_choice => 2),
          Dict(:alg=>QNDF()),
          Dict(:alg=>lsoda()),
          Dict(:alg=>radau()),
          Dict(:alg=>seulex()),
          Dict(:alg=>ImplicitEulerExtrapolation(threading = OrdinaryDiffEq.PolyesterThreads())),
          Dict(:alg=>ImplicitEulerExtrapolation(threading = false)),
          Dict(:alg=>ImplicitEulerBarycentricExtrapolation(min_order = 4, threading = OrdinaryDiffEq.PolyesterThreads())),
          Dict(:alg=>ImplicitEulerBarycentricExtrapolation(min_order = 4, threading = false)),
          Dict(:alg=>ImplicitHairerWannerExtrapolation(threading = OrdinaryDiffEq.PolyesterThreads())),
          Dict(:alg=>ImplicitHairerWannerExtrapolation(threading = false)),
          ]

solnames = ["CVODE_BDF","KenCarp4","Rodas4","Rodas4 Static","Rodas5P","Rodas5P Static","QNDF","lsoda","radau","seulex","ImplEulerExtpl (threaded)", "ImplEulerExtpl (non-threaded)",
           "ImplEulerBaryExtpl (threaded)","ImplEulerBaryExtpl (non-threaded)","ImplHWExtpl (threaded)","ImplHWExtpl (non-threaded)"]

wp = WorkPrecisionSet(probs,abstols,reltols,setups; verbose=false, dense=false,
                      names = solnames,save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)

plot(wp, title = "Implicit Methods: ROBER",legend=:outertopleft,size = (1000,500),
     xticks = 10.0 .^ (-15:1:1),
     yticks = 10.0 .^ (-6:0.3:5),
     bottom_margin= 5Plots.mm)

Error: UndefVarError: `PolyesterThreads` not defined

Conclusion

At high tolerances, Rosenbrock23 and lsoda hit the the error estimates and are fast. At lower tolerances and normal user tolerances, Rodas4 and Rodas5 are extremely fast. lsoda does quite well across both ends. When you get down to reltol=1e-9radau begins to become as efficient as Rodas4, and it continues to do well below that.

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/StiffODE","ROBER.jmd")

Computer Information:

Julia Version 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 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 v0.6.0
⌃ [6e4b80f9] BenchmarkTools v1.5.0
⌃ [f3b72e0c] DiffEqDevTools v2.45.0
⌃ [5b8099bc] DomainSets v0.7.14
  [5a33fad7] GeometricIntegratorsDiffEq v0.2.5
  [40713840] IncompleteLU v0.2.1
  [7f56f5a3] LSODA v0.7.5
⌅ [7ed4a6bd] LinearSolve v2.34.0
⌃ [94925ecb] MethodOfLines v0.11.3
⌃ [961ee093] ModelingToolkit v9.32.0
⌃ [09606e27] ODEInterfaceDiffEq v3.13.3
⌃ [1dea7af3] OrdinaryDiffEq v6.89.0
⌃ [65888b18] ParameterizedFunctions v5.17.0
⌃ [91a5bcdd] Plots v1.40.8
  [132c30aa] ProfileSVG v0.2.2
  [f2c3362d] RecursiveFactorization v0.2.23
  [31c91b34] SciMLBenchmarks v0.1.3
⌃ [90137ffa] StaticArrays v1.9.7
⌃ [c3572dad] Sundials v4.25.0
⌅ [0c5d862f] Symbolics v5.36.0
⌃ [a759f4b9] TimerOutputs v0.5.24
  [37e2e46d] LinearAlgebra
  [2f01184e] SparseArrays v1.10.0
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`
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:

Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/StiffODE/Manifest.toml`
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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 -m`
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