Chemical Akzo Nobel Differential-Algebraic Equation (DAE) Work-Precision Diagrams

using OrdinaryDiffEq, DiffEqDevTools, Sundials, ModelingToolkit, ODEInterfaceDiffEq,
      Plots, DASSL, DASKR
using LinearAlgebra

ModelingToolkit.@parameters begin
      k₁=18.7
      k₂=0.58
      k₃=0.09
      k₄=0.42
      kbig=34.4
      kla=3.3
      ks=115.83
      po2=0.9
      hen=737
end

@variables begin
      t
      y₁(t)  = 0.444
      y₂(t)  = 0.00123
      y₃(t)  = 0.0
      y₄(t)  = 0.007
      y₅(t)  = 1.0
      y₆(t)  = 115.83*0.444*0.007 # ks*y₁*y₄
end

D = Differential(t)

r₁  = k₁ * (y₁^4.)*sqrt(abs(y₂))
r₂  = k₂ * y₃ * y₄
r₃  = k₂/kbig * y₁ * y₅
r₄  = k₃*y₁*(y₄^2)
r₅  = k₄*(y₆^2)*sqrt(abs(y₂))
fin = kla*(po2/hen-y₂)

eqs = [
      D(y₁) ~ -2. * r₁ + r₂ - r₃ - r₄
      D(y₂) ~ -0.5 * r₁ - r₄ - 0.5*r₅ + fin
      D(y₃) ~ r₁ - r₂ + r₃
      D(y₄) ~ -r₂ + r₃ - 2. * r₄
      D(y₅) ~ r₂ - r₃ + r₅
      0. ~ ks * y₁ * y₄ - y₆
]

ModelingToolkit.@named sys = ModelingToolkit.ODESystem(eqs)

tspan = (0.0, 180.0)
mmprob = ODEProblem(sys, [], tspan)
sol = solve(mmprob, Rodas4(),abstol=1/10^14,reltol=1/10^14)

odaeprob = ODAEProblem(structural_simplify(sys),[],tspan)
ode_ref_sol = solve(odaeprob,CVODE_BDF(),abstol=1/10^14,reltol=1/10^14);

du = mmprob.f(mmprob.u0,mmprob.p,0.0)
du0 = D.(states(sys)) .=> du
daeprob = DAEProblem(sys,du0,[],tspan)
ref_sol = solve(daeprob,IDA(),abstol=1/10^14,reltol=1/10^14);

probs = [mmprob,daeprob,odaeprob]
refs = [ref_sol,ref_sol,ode_ref_sol];
plot(ode_ref_sol, vars = [y₁,y₂,y₃,y₄,y₅,y₆])

High Tolerances

abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4);
setups = [Dict(:prob_choice => 1, :alg=>Rosenbrock23()),
          Dict(:prob_choice => 1, :alg=>Rodas4()),
          #Dict(:prob_choice => 1, :alg=>FBDF()),
          Dict(:prob_choice => 1, :alg=>QNDF()),
          Dict(:prob_choice => 1, :alg=>rodas()),
          Dict(:prob_choice => 1, :alg=>radau()),
          Dict(:prob_choice => 1, :alg=>RadauIIA5()),
          Dict(:prob_choice => 2, :alg=>DFBDF()),
          Dict(:prob_choice => 2, :alg=>IDA()),
]

wp = WorkPrecisionSet(probs,abstols,reltols,setups;
                      save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (6:8)
reltols = 1.0 ./ 10.0 .^ (2:4);
setups = [Dict(:prob_choice => 1, :alg=>Rosenbrock23()),
          Dict(:prob_choice => 1, :alg=>Rodas4()),
          Dict(:prob_choice => 2, :alg=>IDA()),
          Dict(:prob_choice => 3, :alg=>Rosenbrock23()),
          Dict(:prob_choice => 3, :alg=>Rodas4()),
          Dict(:prob_choice => 3, :alg=>CVODE_BDF()),
          Dict(:prob_choice => 3, :alg=>TRBDF2()),
          Dict(:prob_choice => 3, :alg=>KenCarp4()),
          ]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
                      save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (6:8)
reltols = 1.0 ./ 10.0 .^ (3:5);
setups = [Dict(:prob_choice => 3, :alg=>Rosenbrock23()),
          Dict(:prob_choice => 3, :alg=>Rodas4()),
          Dict(:prob_choice => 2, :alg=>IDA()),
          Dict(:prob_choice => 2, :alg=>DASSL.dassl()),
          Dict(:prob_choice => 2, :alg=>DASKR.daskr()),
          ]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
                      save_everystep=false,appxsol=refs,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(:prob_choice => 1, :alg=>Rosenbrock23()),
          Dict(:prob_choice => 1, :alg=>Rodas4()),
          #Dict(:prob_choice => 1, :alg=>FBDF()), # too slow
          Dict(:prob_choice => 1, :alg=>QNDF()),
          Dict(:prob_choice => 1, :alg=>rodas()),
          Dict(:prob_choice => 1, :alg=>radau()),
          Dict(:prob_choice => 1, :alg=>RadauIIA5()),
          Dict(:prob_choice => 2, :alg=>DFBDF()),
          Dict(:prob_choice => 2, :alg=>IDA()),
          ]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;error_estimate = :l2,
                      save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (6:8)
reltols = 1.0 ./ 10.0 .^ (2:4);
setups = [Dict(:prob_choice => 1, :alg=>Rosenbrock23()),
          Dict(:prob_choice => 1, :alg=>Rodas4()),
          Dict(:prob_choice => 2, :alg=>IDA()),
          Dict(:prob_choice => 3, :alg=>Rosenbrock23()),
          Dict(:prob_choice => 3, :alg=>Rodas4()),
          Dict(:prob_choice => 3, :alg=>CVODE_BDF()),
          Dict(:prob_choice => 3, :alg=>TRBDF2()),
          Dict(:prob_choice => 3, :alg=>KenCarp4()),
          ]
wp = WorkPrecisionSet(probs,abstols,reltols,setups;error_estimate = :l2,
                      save_everystep=false,appxsol=refs,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(:prob_choice => 1, :alg=>Rodas5()),
          Dict(:prob_choice => 3, :alg=>Rodas5()),
          Dict(:prob_choice => 1, :alg=>Rodas4()),
          Dict(:prob_choice => 3, :alg=>Rodas4()),
          #Dict(:prob_choice => 1, :alg=>FBDF()),
          Dict(:prob_choice => 1, :alg=>QNDF()),
          Dict(:prob_choice => 1, :alg=>rodas()),
          Dict(:prob_choice => 1, :alg=>radau()),
          Dict(:prob_choice => 1, :alg=>RadauIIA5()),
          Dict(:prob_choice => 2, :alg=>DFBDF()),
          Dict(:prob_choice => 2, :alg=>IDA()),
          ]

wp = WorkPrecisionSet(probs,abstols,reltols,setups;
                      save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)

wp = WorkPrecisionSet(probs,abstols,reltols,setups;error_estimate = :l2,
                      save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)

DASKR fails at too low of tolerances, so pull back for a comparison.

abstols = 1.0 ./ 10.0 .^ (7:9)
reltols = 1.0 ./ 10.0 .^ (4:6)

setups = [Dict(:prob_choice => 1, :alg=>Rodas5()),
          Dict(:prob_choice => 3, :alg=>Rodas5()),
          Dict(:prob_choice => 1, :alg=>Rodas4()),
          Dict(:prob_choice => 3, :alg=>Rodas4()),
          Dict(:prob_choice => 2, :alg=>DFBDF()),
          Dict(:prob_choice => 2, :alg=>IDA()),
          Dict(:prob_choice => 2, :alg=>DASKR.daskr()),
          ]
gr()
wp = WorkPrecisionSet(probs,abstols,reltols,setups;
                      save_everystep=false,appxsol=refs,maxiters=Int(1e5),numruns=10)
plot(wp)

Conclusion

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/DAE","ChemicalAkzoNobel.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:
  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-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DAE/Project.toml`
  [165a45c3] DASKR v2.8.0
  [e993076c] DASSL v2.6.0
  [f3b72e0c] DiffEqDevTools v2.31.0
  [961ee093] ModelingToolkit v8.18.0
  [09606e27] ODEInterfaceDiffEq v3.10.1
  [1dea7af3] OrdinaryDiffEq v6.19.2
  [91a5bcdd] Plots v1.31.4
  [31c91b34] SciMLBenchmarks v0.1.0
  [c3572dad] Sundials v4.9.4

And the full manifest:

      Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DAE/Manifest.toml`
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  [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
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  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
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  [f638f0a6] libfdk_aac_jll v2.0.2+0
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  [a9144af2] libsodium_jll v1.0.20+0
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  [76f85450] LibGit2
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  [a4e569a6] Tar v1.10.0
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  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.0+0
  [14a3606d] MozillaCACerts_jll v2022.2.1
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  [05823500] OpenLibm_jll v0.8.1+0
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  [bea87d4a] SuiteSparse_jll v5.10.1+0
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  [8e850b90] libblastrampoline_jll v5.1.0+0
  [8e850ede] nghttp2_jll v1.41.0+1
  [3f19e933] p7zip_jll v17.4.0+0