# Wheldon, Kirk, and Finlay

We study algorithms for solving constant delay differential equations with a test problem from W.H. Enright and H. Hayashi, "The evaluation of numerical software for delay differential equations", 1997. It is a model of chronic granulocytic leukemia that was published by T. Wheldon, J. Kirk and H. Finlay in "Cyclical granulopoiesis in chronic granulocytic leukemia: A simulation study", 1974, and is given by

\begin{align} y_1'(t) &= \frac{1.1}{1 + \sqrt{10}y_1(t-20)^{5/4}} - \frac{10y_1(t)}{1 + 40y_2(t)} \\ y_2'(t) &= \frac{100y_1(t)}{1 + 40y_2(t)} - 2.43y_2(t) \end{align}
using DelayDiffEq, DiffEqDevTools, DiffEqProblemLibrary, Plots
using DiffEqProblemLibrary.DDEProblemLibrary: importddeproblems; importddeproblems()
import DiffEqProblemLibrary.DDEProblemLibrary: prob_dde_wheldon
gr()

sol = solve(prob_dde_wheldon, MethodOfSteps(Vern9(); fpsolve = NLFunctional(; max_iter = 1000)); reltol=1e-14, abstol=1e-14)
test_sol = TestSolution(sol)
plot(sol)

## Low order RK methods

### High tolerances

First we compare final errors of solutions with low order RK methods at high tolerances.

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(BS3())),
Dict(:alg=>MethodOfSteps(Tsit5())),
Dict(:alg=>MethodOfSteps(RK4())),
Dict(:alg=>MethodOfSteps(DP5())),
Dict(:alg=>MethodOfSteps(OwrenZen3())),
Dict(:alg=>MethodOfSteps(OwrenZen4())),
Dict(:alg=>MethodOfSteps(OwrenZen5()))]
names = ["BS3", "Tsit5", "RK4", "DP5", "OwrenZen3", "OwrenZen4", "OwrenZen5"]
wp = WorkPrecisionSet(prob_dde_wheldon,abstols,reltols,setups;names=names,
appxsol=test_sol,maxiters=Int(1e5),error_estimate=:final)
plot(wp)

Next we test interpolation errors:

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(BS3())),
Dict(:alg=>MethodOfSteps(Tsit5())),
Dict(:alg=>MethodOfSteps(RK4())),
Dict(:alg=>MethodOfSteps(DP5())),
Dict(:alg=>MethodOfSteps(OwrenZen3())),
Dict(:alg=>MethodOfSteps(OwrenZen4())),
Dict(:alg=>MethodOfSteps(OwrenZen5()))]
names = ["BS3", "Tsit5", "RK4", "DP5", "OwrenZen3", "OwrenZen4", "OwrenZen5"]
wp = WorkPrecisionSet(prob_dde_wheldon,abstols,reltols,setups;names=names,
appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

Both interpolation tests and tests of final error show similar results. BS3 does quite well but only OwrenZen4, OwrenZen5, and RK4 achieve interpolation errors of about 1e-5.

### Low tolerances

We repeat our tests at low tolerances.

abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)

setups = [Dict(:alg=>MethodOfSteps(BS3())),
Dict(:alg=>MethodOfSteps(Tsit5())),
Dict(:alg=>MethodOfSteps(RK4())),
Dict(:alg=>MethodOfSteps(DP5())),
Dict(:alg=>MethodOfSteps(OwrenZen3())),
Dict(:alg=>MethodOfSteps(OwrenZen4())),
Dict(:alg=>MethodOfSteps(OwrenZen5()))]
names = ["BS3", "Tsit5", "RK4", "DP5", "OwrenZen3", "OwrenZen4", "OwrenZen5"]
wp = WorkPrecisionSet(prob_dde_wheldon,abstols,reltols,setups;names=names,
appxsol=test_sol,maxiters=Int(1e5),error_estimate=:final)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)

setups = [Dict(:alg=>MethodOfSteps(BS3())),
Dict(:alg=>MethodOfSteps(Tsit5())),
Dict(:alg=>MethodOfSteps(RK4())),
Dict(:alg=>MethodOfSteps(DP5())),
Dict(:alg=>MethodOfSteps(OwrenZen3())),
Dict(:alg=>MethodOfSteps(OwrenZen4())),
Dict(:alg=>MethodOfSteps(OwrenZen5()))]
names = ["BS3", "Tsit5", "RK4", "DP5", "OwrenZen3", "OwrenZen4", "OwrenZen5"]
wp = WorkPrecisionSet(prob_dde_wheldon,abstols,reltols,setups;names=names,
appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

Out of the compared methods, Tsit5, DP5, and OwrenZen5 seem to be the best methods for this problem at low tolerances, but also OwrenZen4 performs similarly well. OwrenZen5 and OwrenZen4 can even achieve interpolation errors below 1e-9.

## Lazy interpolants

### High tolerances

We compare the Verner methods, which use lazy interpolants, at high tolerances. As reference we include OwrenZen4.

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(Vern6())),
Dict(:alg=>MethodOfSteps(Vern7())),
Dict(:alg=>MethodOfSteps(Vern8())),
Dict(:alg=>MethodOfSteps(Vern9())),
Dict(:alg=>MethodOfSteps(OwrenZen4()))]
names = ["Vern6", "Vern7", "Vern8", "Vern9", "OwrenZen4"]
wp = WorkPrecisionSet(prob_dde_wheldon,abstols,reltols,setups;names=names,
appxsol=test_sol,maxiters=Int(1e5),error_estimate=:final)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)

setups = [Dict(:alg=>MethodOfSteps(Vern6())),
Dict(:alg=>MethodOfSteps(Vern7())),
Dict(:alg=>MethodOfSteps(Vern8())),
Dict(:alg=>MethodOfSteps(Vern9())),
Dict(:alg=>MethodOfSteps(OwrenZen4()))]
names = ["Vern6", "Vern7", "Vern8", "Vern9", "OwrenZen4"]
wp = WorkPrecisionSet(prob_dde_wheldon,abstols,reltols,setups;names=names,
appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

### Low tolerances

We repeat these tests and compare the Verner methods also at low tolerances.

abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)

setups = [Dict(:alg=>MethodOfSteps(Vern6())),
Dict(:alg=>MethodOfSteps(Vern7())),
Dict(:alg=>MethodOfSteps(Vern8())),
Dict(:alg=>MethodOfSteps(Vern9())),
Dict(:alg=>MethodOfSteps(OwrenZen4()))]
names = ["Vern6", "Vern7", "Vern8", "Vern9", "OwrenZen4"]
wp = WorkPrecisionSet(prob_dde_wheldon,abstols,reltols,setups;names=names,
appxsol=test_sol,maxiters=Int(1e5),error_estimate=:final)
plot(wp)

abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)

setups = [Dict(:alg=>MethodOfSteps(Vern6())),
Dict(:alg=>MethodOfSteps(Vern7())),
Dict(:alg=>MethodOfSteps(Vern8())),
Dict(:alg=>MethodOfSteps(Vern9())),
Dict(:alg=>MethodOfSteps(OwrenZen4()))]
names = ["Vern6", "Vern7", "Vern8", "Vern9", "OwrenZen4"]
wp = WorkPrecisionSet(prob_dde_wheldon,abstols,reltols,setups;names=names,
appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

It seems Vern6 and Vern7 are both well suited for the problem at low tolerances and outperform OwrenZen4, whereas at high tolerances OwrenZen4 is more efficient.

## 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/NonStiffDDE","Wheldon_Kirk_Finlay_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:
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/NonStiffDDE/Project.toml
[bcd4f6db] DelayDiffEq v5.37.0
[f3b72e0c] DiffEqDevTools v2.30.0
[a077e3f3] DiffEqProblemLibrary v4.17.0
[91a5bcdd] Plots v1.31.4
[31c91b34] SciMLBenchmarks v0.1.0

And the full manifest:

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[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
[35661453] Xorg_xkbcomp_jll v1.4.2+4
[33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
[c5fb5394] Xorg_xtrans_jll v1.4.0+3
[8f1865be] ZeroMQ_jll v4.3.4+0
[3161d3a3] Zstd_jll v1.5.2+0
[a4ae2306] libaom_jll v3.4.0+0
[0ac62f75] libass_jll v0.15.1+0
[f638f0a6] libfdk_aac_jll v2.0.2+0
[b53b4c65] libpng_jll v1.6.38+0
[a9144af2] libsodium_jll v1.0.20+0
[f27f6e37] libvorbis_jll v1.3.7+1
[1270edf5] x264_jll v2021.5.5+0
[dfaa095f] x265_jll v3.5.0+0
[d8fb68d0] xkbcommon_jll v0.9.1+5
[56f22d72] Artifacts
[2a0f44e3] Base64
[8bb1440f] DelimitedFiles
[8ba89e20] Distributed
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[b27032c2] LibCURL v0.6.3
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.8.0
[de0858da] Printf
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays
[10745b16] Statistics
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.0
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v0.5.2+0
[deac9b47] LibCURL_jll v7.81.0+0
[29816b5a] LibSSH2_jll v1.10.2+0
[c8ffd9c3] MbedTLS_jll v2.28.0+0
[14a3606d] MozillaCACerts_jll v2022.2.1
[4536629a] OpenBLAS_jll v0.3.20+0
[05823500] OpenLibm_jll v0.8.1+0
[efcefdf7] PCRE2_jll v10.40.0+0
[bea87d4a] SuiteSparse_jll v5.10.1+0
[83775a58] Zlib_jll v1.2.12+3
[8e850b90] libblastrampoline_jll v5.1.0+0
[8e850ede] nghttp2_jll v1.41.0+1
[3f19e933] p7zip_jll v17.4.0+0