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RADAR5 Oregonator Work-Precision Diagrams

RADAR5 Oregonator

This is a stiff delay differential equation model from chemical kinetics, taken from the RADAR5 test suite by Guglielmi and Hairer. The problem is given by

\[u_1'(t) = k_1 A u_2(t) - k_2 u_1(t) u_2(t - \tau) + k_3 B u_1(t) - 2 k_4 u_1(t)^2\]

\[u_2'(t) = -k_1 A u_2(t) - k_2 u_1(t) u_2(t - \tau) + f k_3 B u_1(t)\]

for $t \in [0, 100.5]$ with history function $\phi_1(t) = 10^{-10}$, $\phi_2(t) = 10^{-5}$ for $t \leq 0$, where $k_1 = 1.34$, $k_2 = 1.6 \times 10^9$, $k_3 = 8000$, $k_4 = 4 \times 10^7$, $f = 1$, $A = 0.06$, $B = 0.06$, and $\tau = 0.15$.

This problem is extremely stiff due to the large rate constants ($k_2 = 1.6 \times 10^9$, $k_4 = 4 \times 10^7$).

References

Epstein, I. and Luo, Y. (1991). Differential delay equations in chemical kinetics. Nonlinear models, Journal of Chemical Physics (95), pp. 244-254.

Guglielmi, N. and Hairer, E. (2001). Implementing Radau IIA methods for stiff delay differential equations, Computing (67), pp. 1-12.

using DelayDiffEq, DiffEqDevTools, DDEProblemLibrary, Plots
using OrdinaryDiffEqLowOrderRK, OrdinaryDiffEqRosenbrock, OrdinaryDiffEqSDIRK,
    OrdinaryDiffEqTsit5
import DDEProblemLibrary: prob_dde_RADAR5_oregonator
gr()
Plots.GRBackend()

Reference Solution

We compute a reference solution using Rodas5P at very tight tolerances. The solution components range from about $10^{-11}$ to $10^{-5}$, so absolute tolerances are scaled with the solution as in the RADAR5 driver (ATOL = RTOL * 1e-9). After an initial decay of $u_2$, the solution settles into periodic relaxation spikes.

sol = solve(prob_dde_RADAR5_oregonator, MethodOfSteps(Rodas5P());
    reltol = 1e-14, abstol = 1e-21)
test_sol = TestSolution(sol)
plot(sol; title = "RADAR5 Oregonator Solution")

High Tolerances

Rosenbrock methods

abstols = 1.0 ./ 10.0 .^ (10:13)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg => MethodOfSteps(Rosenbrock23())),
    Dict(:alg => MethodOfSteps(Rodas4())),
    Dict(:alg => MethodOfSteps(Rodas5())),
    Dict(:alg => MethodOfSteps(Rodas5P()))]
names = ["Rosenbrock23", "Rodas4", "Rodas5", "Rodas5P"]
wp = WorkPrecisionSet(prob_dde_RADAR5_oregonator, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :final)
plot(wp; title = "Oregonator: Rosenbrock Methods (final error)")

wp = WorkPrecisionSet(prob_dde_RADAR5_oregonator, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L2)
plot(wp; title = "Oregonator: Rosenbrock Methods (L2 error)")

wp = WorkPrecisionSet(prob_dde_RADAR5_oregonator, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L∞)
plot(wp; title = "Oregonator: Rosenbrock Methods (L∞ error)")

SDIRK methods

At the loosest high-tolerance pair (abstol = 1e-10, reltol = 1e-1), TRBDF2 returns Unstable before $t = 100.5$, so that point is omitted from the diagrams. Final errors for the SDIRK methods stay between about $2\times 10^{-9}$ and $4\times 10^{-8}$ across the high-tolerance range and are not monotone in the tolerance; on this relaxation oscillator the final-error estimate is phase-sensitive, so tightening the tolerance does not always improve it.

setups = [Dict(:alg => MethodOfSteps(TRBDF2())),
    Dict(:alg => MethodOfSteps(SDIRK2())),
    Dict(:alg => MethodOfSteps(KenCarp4())),
    Dict(:alg => MethodOfSteps(Kvaerno4())),
    Dict(:alg => MethodOfSteps(Kvaerno5()))]
names = ["TRBDF2", "SDIRK2", "KenCarp4", "Kvaerno4", "Kvaerno5"]
wp = WorkPrecisionSet(prob_dde_RADAR5_oregonator, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :final)
plot(wp; title = "Oregonator: SDIRK Methods (final error)")

wp = WorkPrecisionSet(prob_dde_RADAR5_oregonator, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L2)
plot(wp; title = "Oregonator: SDIRK Methods (L2 error)")

wp = WorkPrecisionSet(prob_dde_RADAR5_oregonator, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L∞)
plot(wp; title = "Oregonator: SDIRK Methods (L∞ error)")

Stiff vs Non-Stiff Comparison

In the latest run, Tsit5 and DP5 are more expensive than Rosenbrock23 and Rodas5P at every error they reach. BS3 costs about as much as Rosenbrock23 at the smallest errors. At the loosest high-tolerance pair, Tsit5 and BS3 return Unstable (near $t \approx 68$ and $t \approx 28$), and TRBDF2 fails the same way as in the SDIRK diagram; those points are omitted. DP5 finishes but with a final error of about $1.5\times 10^{-7}$, roughly 40% of $u_2$ at $t_{\mathrm{end}}$ ($\approx 3.6\times 10^{-7}$), making it the least accurate point in the diagram.

setups = [Dict(:alg => MethodOfSteps(Rosenbrock23())),
    Dict(:alg => MethodOfSteps(Rodas5P())),
    Dict(:alg => MethodOfSteps(TRBDF2())),
    Dict(:alg => MethodOfSteps(KenCarp4())),
    Dict(:alg => MethodOfSteps(Tsit5())),
    Dict(:alg => MethodOfSteps(BS3())),
    Dict(:alg => MethodOfSteps(DP5()))]
names = ["Rosenbrock23", "Rodas5P", "TRBDF2", "KenCarp4", "Tsit5", "BS3", "DP5"]
wp = WorkPrecisionSet(prob_dde_RADAR5_oregonator, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e6), error_estimate = :final)
plot(wp; title = "Oregonator: Stiff vs Non-Stiff (final error)")

Low Tolerances

At these tolerances Rosenbrock23 needs about $7\times 10^4$ to $2.4\times 10^6$ accepted steps per solve, so this block raises maxiters to Int(1e7). The Rodas and SDIRK methods below complete within $10^5$ iterations.

abstols = 1.0 ./ 10.0 .^ (14:17)
reltols = 1.0 ./ 10.0 .^ (5:8)

setups = [Dict(:alg => MethodOfSteps(Rosenbrock23())),
    Dict(:alg => MethodOfSteps(Rodas4())),
    Dict(:alg => MethodOfSteps(Rodas5())),
    Dict(:alg => MethodOfSteps(Rodas5P())),
    Dict(:alg => MethodOfSteps(TRBDF2())),
    Dict(:alg => MethodOfSteps(KenCarp4()))]
names = ["Rosenbrock23", "Rodas4", "Rodas5", "Rodas5P", "TRBDF2", "KenCarp4"]
wp = WorkPrecisionSet(prob_dde_RADAR5_oregonator, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e7), error_estimate = :final)
plot(wp; title = "Oregonator: Low Tolerances (final error)")

wp = WorkPrecisionSet(prob_dde_RADAR5_oregonator, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e7), error_estimate = :L2)
plot(wp; title = "Oregonator: Low Tolerances (L2 error)")

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/StiffDDE","Oregonator_wpd.jmd")

Computer Information:

Julia Version 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 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
  LLVM: libLLVM-18.1.7 (ORCJIT, znver2)
  GC: Built with stock GC
Threads: 128 default, 1 interactive, 128 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

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Info Packages marked with ⌃ have new versions available and may be upgradable.

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  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
⌃ [a4ae2306] libaom_jll v3.14.1+0
  [0ac62f75] libass_jll v0.17.5+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [8e53e030] libdrm_jll v2.4.134+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
⌃ [b53b4c65] libpng_jll v1.6.58+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.7.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.3.0
  [44cfe95a] Pkg v1.12.1
  [de0858da] Printf v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.12.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.3.1+2
  [deac9b47] LibCURL_jll v8.15.0+0
  [e37daf67] LibGit2_jll v1.9.0+0
  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.11.4
  [4536629a] OpenBLAS_jll v0.3.29+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.6+0
  [efcefdf7] PCRE2_jll v10.44.0+1
  [bea87d4a] SuiteSparse_jll v7.8.3+2
  [83775a58] Zlib_jll v1.3.1+2
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.64.0+1
  [3f19e933] p7zip_jll v17.7.0+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 -m`