RADAR5 Robertson Work-Precision Diagrams

RADAR5 Robertson Chemical Reaction

This is a stiff delay differential equation model of a chemical reaction with a steady state solution, taken from the RADAR5 test suite by Guglielmi and Hairer. The system has dimension 3 with one constant delay.

\[u_1'(t) = -a \, u_1(t) + b \, u_2(t - \tau) \, u_3(t)\]

\[u_2'(t) = a \, u_1(t) - b \, u_2(t - \tau) \, u_3(t) - c \, u_2(t)^2\]

\[u_3'(t) = c \, u_2(t)^2\]

for $t \in [0, 1]$ with initial conditions $u_1(0) = 1$, $u_2(t) = 0$ for $t \in [-\tau, 0]$, and $u_3(0) = 0$, where $a = 0.04$, $b = 10000$, $c = 3 \times 10^7$, and $\tau = 0.01$.

This is the DDE analogue of the classic Robertson ODE problem. The extreme stiffness arises from the ratio $c/a = 7.5 \times 10^8$ between the fastest and slowest time scales. In the original RADAR5 formulation the problem is integrated up to $t = 10^{10}$, but this is currently not achievable with the method of steps approach in DelayDiffEq.jl. We benchmark on the interval $[0, 1]$ which already demonstrates the stiffness challenges.

References

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 OrdinaryDiffEqCore: IController
using OrdinaryDiffEqLowOrderRK, OrdinaryDiffEqRosenbrock, OrdinaryDiffEqSDIRK,
    OrdinaryDiffEqTsit5
import DDEProblemLibrary: prob_dde_RADAR5_robertson
gr()

Plots.GRBackend()

Problem Setup

We use a truncated time span of $[0, 1]$ for benchmarking:

prob = remake(prob_dde_RADAR5_robertson; tspan = (0.0, 1.0))

DDEProblem with uType Vector{Float64} and tType Float64. In-place: true
Non-trivial mass matrix: false
timespan: (0.0, 1.0)
u0: 3-element Vector{Float64}:
 1.0
 0.0
 0.0

Reference Solution

Due to the extreme stiffness and the small scale of the second component ($u_2 \sim O(10^{-5})$), we use component-wise absolute tolerances to compute the reference solution:

sol = solve(prob, MethodOfSteps(Rodas5P());
    reltol = 1e-14, abstol = [1e-14, 1e-20, 1e-14], dt = 1e-6)
test_sol = TestSolution(sol)
plot(sol; title = "Robertson DDE Solution (t ∈ [0, 1])")

High Tolerances

Rosenbrock methods

abstols = 1.0 ./ 10.0 .^ (4:7)
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, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :final,
    dt = 1e-6)
plot(wp; title = "Robertson: Rosenbrock Methods (final error)")

wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L2,
    dt = 1e-6)
plot(wp; title = "Robertson: Rosenbrock Methods (L2 error)")

SDIRK methods

setups = [Dict(:alg => MethodOfSteps(TRBDF2())),
    Dict(:alg => MethodOfSteps(SDIRK2()),
        :controller => IController(SDIRK2(); qmax = 10, qmax_first_step = 10)),
    Dict(:alg => MethodOfSteps(KenCarp4()))]
names = ["TRBDF2", "SDIRK2", "KenCarp4"]
wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :final,
    dt = 1e-6)
plot(wp; title = "Robertson: SDIRK Methods (final error)")

wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L2,
    dt = 1e-6)
plot(wp; title = "Robertson: SDIRK Methods (L2 error)")

Stiff vs Non-Stiff Comparison

setups = [Dict(:alg => MethodOfSteps(Rodas5P())),
    Dict(:alg => MethodOfSteps(TRBDF2())),
    Dict(:alg => MethodOfSteps(KenCarp4())),
    Dict(:alg => MethodOfSteps(Tsit5())),
    Dict(:alg => MethodOfSteps(DP5()))]
names = ["Rodas5P", "TRBDF2", "KenCarp4", "Tsit5", "DP5"]
wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e6), error_estimate = :final,
    dt = 1e-6)
plot(wp; title = "Robertson: Stiff vs Non-Stiff (final error)")

Low Tolerances

abstols = 1.0 ./ 10.0 .^ (8:11)
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, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :final,
    dt = 1e-6)
plot(wp; title = "Robertson: Low Tolerances (final error)")

wp = WorkPrecisionSet(prob, abstols, reltols, setups;
    names = names, appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L2,
    dt = 1e-6)
plot(wp; title = "Robertson: 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","Robertson_wpd.jmd")

Computer Information:

Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 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-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/StiffDDE/Project.toml`
  [f42792ee] DDEProblemLibrary v0.1.9
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⌃ [31c91b34] SciMLBenchmarks v0.1.3
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

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  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [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 v1.11.0
  [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 v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
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`