Quadrupole Boson Energy Conservation
The Quadrupole Boson Hamiltonian is a 2D Hamiltonian system with cubic and quartic potential terms. This benchmark compares the energy preservation of different solver approaches using a work-precision diagram.
Code:
using LinearAlgebra, Statisticsusing SciMLBase, OrdinaryDiffEq, DiffEqCallbacks, Plotsusing ProbNumDiffEqPlots.theme( :dao; markerstrokewidth=0.5, legend=:outertopright, margin=5Plots.mm, xticks=10.0 .^ (-16:1:16),)Code:
const A_QB = 1.0const B_QB = 0.55const D_QB = 0.4function QB_acceleration!(ddu, du, u, p, t) q1, q2 = u r2 = q1^2 + q2^2 ddu[1] = -A_QB * q1 - 3 * B_QB / sqrt(2) * (q2^2 - q1^2) - D_QB * q1 * r2 ddu[2] = -q2 * (A_QB + 3 * sqrt(2) * B_QB * q1 + D_QB * r2)endu0 = [4.919080920016389, 2.836942666663649]du0 = [0.0, 0.0]tspan = (0.0, 1000.0)prob_2nd = SecondOrderODEProblem(QB_acceleration!, du0, u0, tspan)function QB_dp!(dp, p, q, params, t) q1, q2 = q r2 = q1^2 + q2^2 dp[1] = -A_QB * q1 - 3 * B_QB / sqrt(2) * (q2^2 - q1^2) - D_QB * q1 * r2 dp[2] = -q2 * (A_QB + 3 * sqrt(2) * B_QB * q1 + D_QB * r2)endfunction QB_dq!(dq, p, q, params, t) dq[1] = A_QB * p[1] dq[2] = A_QB * p[2]endprob_dyn = DynamicalODEProblem(QB_dp!, QB_dq!, du0, u0, tspan)function H_QB(dx, dy, x, y) r2 = x^2 + y^2 A_QB / 2 * (dx^2 + dy^2) + A_QB / 2 * r2 + B_QB / sqrt(2) * x * (3 * y^2 - x^2) + D_QB / 4 * r2^2endH_QB(u) = H_QB(u...)E0 = H_QB(du0..., u0...)ref_sol = solve(prob_2nd, Vern9(), abstol=1/10^14, reltol=1/10^14)plot(ref_sol, idxs=(3, 4), title="Quadrupole Boson Solution", legend=false, xticks=:auto, yticks=:auto)
Energy Error Work-Precision Diagram
Code:
# `ManifoldUpdate` fires as a `DiscreteCallback`, which by default saves the state both# before and after its `affect!`; since `savevalues!` at the "before" position always# runs regardless of `save_positions` when `save_everystep=true`, both the raw and the# manifold-corrected state end up in `sol.u` at the same `t`. Keep only the last (i.e.# manifold-corrected) value per unique `t` so the error metric reflects the actual output.function last_per_t(sol) n = length(sol.t) return (sol.u[i] for i in 1:n if i == n || sol.t[i] != sol.t[i+1])endfunction adaptive_energy_wpd(prob, alg, abstols, reltols, E0, Hfunc; numruns=5, kwargs...) errors = Float64[] times = Float64[] nevals = Int[] for (abstol, reltol) in zip(abstols, reltols) kw = (; abstol, reltol, dense=false, maxiters=Int(1e7), kwargs...) local sol try sol = solve(prob, alg; kw...) catch e @warn "solve failed, skipping" alg abstol reltol exception = e continue end sol.retcode == SciMLBase.ReturnCode.Success || continue push!(errors, maximum(abs(Hfunc(u) - E0) for u in last_per_t(sol))) push!(nevals, sol.stats.nf + sol.stats.nf2) solve(prob, alg; kw...) t = minimum(@elapsed(solve(prob, alg; kw...)) for _ in 1:numruns) push!(times, t) end errors, times, nevalsendfunction fixedstep_energy_wpd(prob, alg, dts, E0, Hfunc; numruns=5, kwargs...) errors = Float64[] times = Float64[] nevals = Int[] for dt in dts kw = (; dt, adaptive=false, dense=false, maxiters=Int(1e7), kwargs...) local sol try sol = solve(prob, alg; kw...) catch e @warn "solve failed, skipping" alg dt exception = e continue end sol.retcode == SciMLBase.ReturnCode.Success || continue push!(errors, maximum(abs(Hfunc(u) - E0) for u in last_per_t(sol))) push!(nevals, sol.stats.nf + sol.stats.nf2) solve(prob, alg; kw...) t = minimum(@elapsed(solve(prob, alg; kw...)) for _ in 1:numruns) push!(times, t) end errors, times, nevalsendCode:
abstols = 1.0 ./ 10.0 .^ (5:11)reltols = 1.0 ./ 10.0 .^ (2:8)dts = 10.0 .^ range(-0.5, -2.5, length=7)residual(u) = [H_QB(u) - E0]cb_mu = ManifoldUpdate(residual)function energy_manifold!(resid, u, p) resid[1] = H_QB(u) - E0endfunction energy_jacobian!(J, u, p) dx, dy, x, y = u r2 = x^2 + y^2 J[1, 1] = A_QB * dx J[1, 2] = A_QB * dy J[1, 3] = A_QB * x + B_QB / sqrt(2) * (3 * y^2 - 3 * x^2) + D_QB * x * r2 J[1, 4] = A_QB * y + 6 * B_QB / sqrt(2) * x * y + D_QB * y * r2endcb_mp = ManifoldProjection(energy_manifold!; manifold_jacobian=energy_jacobian!, resid_prototype=zeros(1), autonomous=Val(true))Code:
e_ek1_mu3, t_ek1_mu3, n_ek1_mu3 = adaptive_energy_wpd( prob_2nd, EK1(order=3, smooth=false), abstols, reltols, E0, H_QB; callback=cb_mu)e_ek1_mu5, t_ek1_mu5, n_ek1_mu5 = adaptive_energy_wpd( prob_2nd, EK1(order=5, smooth=false), abstols, reltols, E0, H_QB; callback=cb_mu)e_ek1_mu8, t_ek1_mu8, n_ek1_mu8 = adaptive_energy_wpd( prob_2nd, EK1(order=8, smooth=false), abstols, reltols, E0, H_QB; callback=cb_mu)e_ek1_3, t_ek1_3, n_ek1_3 = adaptive_energy_wpd( prob_2nd, EK1(order=3, smooth=false), abstols, reltols, E0, H_QB)e_ek1_5, t_ek1_5, n_ek1_5 = adaptive_energy_wpd( prob_2nd, EK1(order=5, smooth=false), abstols, reltols, E0, H_QB)e_ek1_8, t_ek1_8, n_ek1_8 = adaptive_energy_wpd( prob_2nd, EK1(order=8, smooth=false), abstols, reltols, E0, H_QB)e_kl8, t_kl8, n_kl8 = fixedstep_energy_wpd(prob_dyn, KahanLi8(), dts, E0, H_QB)e_ss10, t_ss10, n_ss10 = fixedstep_energy_wpd(prob_dyn, SofSpa10(), dts, E0, H_QB)e_dprkn6, t_dprkn6, n_dprkn6 = adaptive_energy_wpd(prob_2nd, DPRKN6(), abstols, reltols, E0, H_QB)e_vern9_mp, t_vern9_mp, n_vern9_mp = adaptive_energy_wpd( prob_2nd, Vern9(), abstols, reltols, E0, H_QB; callback=cb_mp)e_vern9, t_vern9, n_vern9 = adaptive_energy_wpd(prob_2nd, Vern9(), abstols, reltols, E0, H_QB)Code:
plot(xlabel="Max. Energy Error", ylabel="Time (s)", xscale=:log10, yscale=:log10, title="Quadrupole Boson Energy Conservation (tspan=1000)")plot!(e_ek1_mu3, t_ek1_mu3, label="EK1(3) + ManifoldUpdate", color=1, marker=:circle, linestyle=:dashdot)plot!(e_ek1_mu5, t_ek1_mu5, label="EK1(5) + ManifoldUpdate", color=1, marker=:circle)plot!(e_ek1_mu8, t_ek1_mu8, label="EK1(8) + ManifoldUpdate", color=1, marker=:circle, linestyle=:dash)plot!(e_ek1_3, t_ek1_3, label="EK1(3)", color=2, marker=:diamond, linestyle=:dashdot)plot!(e_ek1_5, t_ek1_5, label="EK1(5)", color=2, marker=:diamond)plot!(e_ek1_8, t_ek1_8, label="EK1(8)", color=2, marker=:diamond, linestyle=:dash)plot!(e_kl8, t_kl8, label="KahanLi8", color=:gray, marker=:star5)plot!(e_ss10, t_ss10, label="SofSpa10", color=:gray, marker=:star5, linestyle=:dash)plot!(e_dprkn6, t_dprkn6, label="DPRKN6", color=:gray, marker=:utriangle, linestyle=:dot)plot!(e_vern9_mp, t_vern9_mp, label="Vern9 + ManifoldProjection", color=:gray, marker=:square)plot!(e_vern9, t_vern9, label="Vern9", color=:gray, marker=:square, linestyle=:dash)
Code:
plot(xlabel="Max. Energy Error", ylabel="Number of f Evaluations", xscale=:log10, yscale=:log10, title="Quadrupole Boson Energy Conservation (tspan=1000)")plot!(e_ek1_mu3, n_ek1_mu3, label="EK1(3) + ManifoldUpdate", color=1, marker=:circle, linestyle=:dashdot)plot!(e_ek1_mu5, n_ek1_mu5, label="EK1(5) + ManifoldUpdate", color=1, marker=:circle)plot!(e_ek1_mu8, n_ek1_mu8, label="EK1(8) + ManifoldUpdate", color=1, marker=:circle, linestyle=:dash)plot!(e_ek1_3, n_ek1_3, label="EK1(3)", color=2, marker=:diamond, linestyle=:dashdot)plot!(e_ek1_5, n_ek1_5, label="EK1(5)", color=2, marker=:diamond)plot!(e_ek1_8, n_ek1_8, label="EK1(8)", color=2, marker=:diamond, linestyle=:dash)plot!(e_kl8, n_kl8, label="KahanLi8", color=:gray, marker=:star5)plot!(e_ss10, n_ss10, label="SofSpa10", color=:gray, marker=:star5, linestyle=:dash)plot!(e_dprkn6, n_dprkn6, label="DPRKN6", color=:gray, marker=:utriangle, linestyle=:dot)plot!(e_vern9_mp, n_vern9_mp, label="Vern9 + ManifoldProjection", color=:gray, marker=:square)plot!(e_vern9, n_vern9, label="Vern9", color=:gray, marker=:square, linestyle=:dash)
Appendix
Computer information:
using InteractiveUtilsInteractiveUtils.versioninfo()Julia Version 1.13.0
Commit d1c37793dd2 (2026-09-09 19:00 UTC)
Build Info:
Official https://julialang.org release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 128 × AMD Ryzen Threadripper PRO 7985WX 64-Cores
WORD_SIZE: 64
LLVM: libLLVM-20.1.8 (ORCJIT, znver4)
GC: Built with stock GC
Threads: 16 default, 1 interactive, 16 GC (on 128 virtual cores)
Environment:
LD_LIBRARY_PATH = Package information:
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[05181044] RelocatableFolders v1.0.1
[ae029012] Requires v1.3.1
[ae5879a3] ResettableStacks v1.4.0
[79098fc4] Rmath v0.9.0
[47965b36] RootedTrees v2.27.0
[f2b01f46] Roots v3.0.8
[7e49a35a] RuntimeGeneratedFunctions v0.5.26
⌃ [9dfe8606] SCCNonlinearSolve v1.13.0
[94e857df] SIMDTypes v0.1.0
[476501e8] SLEEFPirates v0.6.46
⌅ [0bca4576] SciMLBase v2.155.2
⌃ [19f34311] SciMLJacobianOperators v0.1.17
⌅ [a6db7da4] SciMLLogging v1.10.1
[c0aeaf25] SciMLOperators v1.30.1
[431bcebd] SciMLPublic v1.3.0
[53ae85a6] SciMLStructures v1.10.5
⌃ [505e40e9] SciPyDiffEq v0.2.9
[7e506255] ScopedValues v1.6.2
[6c6a2e73] Scratch v1.3.0
[efcf1570] Setfield v1.1.2
[992d4aef] Showoff v1.1.1
⌃ [727e6d20] SimpleNonlinearSolve v2.12.0
[699a6c99] SimpleTraits v0.9.6
[ce78b400] SimpleUnPack v1.1.0
[a2af1166] SortingAlgorithms v1.2.3
[a57abbd0] SparseColumnPivotedQR v2.1.8
[0a514795] SparseMatrixColorings v0.4.28
[276daf66] SpecialFunctions v2.9.0
[860ef19b] StableRNGs v1.0.4
[0c0c59c1] StarAlgebras v0.3.0
[64909d44] StateSelection v1.11.1
[aedffcd0] Static v1.4.6
[0d7ed370] StaticArrayInterface v1.10.0
[90137ffa] StaticArrays v1.9.20
[1e83bf80] StaticArraysCore v1.4.4
[10745b16] Statistics v1.11.5
[82ae8749] StatsAPI v1.8.0
[2913bbd2] StatsBase v0.34.13
[4c63d2b9] StatsFuns v2.2.1
[7792a7ef] StrideArraysCore v0.5.9
[69024149] StringEncodings v0.3.7
[09ab397b] StructArrays v0.7.3
⌃ [c3572dad] Sundials v6.3.0
[2efcf032] SymbolicIndexingInterface v0.3.55
[19f23fe9] SymbolicLimits v1.2.1
⌅ [d1185830] SymbolicUtils v4.45.0
⌃ [0c5d862f] Symbolics v7.39.0
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.14.0
[ed4db957] TaskLocalValues v0.1.3
⌃ [92b13dbe] TaylorIntegration v0.18.14
[6aa5eb33] TaylorSeries v0.22.6
[62fd8b95] TensorCore v0.1.1
[8ea1fca8] TermInterface v2.0.0
[8290d209] ThreadingUtilities v0.5.6
⌅ [a759f4b9] TimerOutputs v0.5.29
[c751599d] ToeplitzMatrices v0.8.5
[781d530d] TruncatedStacktraces v1.4.0
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
[41fe7b60] Unzip v0.2.0
[3d5dd08c] VectorizationBase v0.21.74
[81def892] VersionParsing v1.3.0
[d30d5f5c] WeakCacheSets v0.1.0
[44d3d7a6] Weave v0.10.12
[ddb6d928] YAML v0.4.16
[c2297ded] ZMQ v1.5.1
⌃ [0518478a] deSolveDiffEq v1.4.1
[6e34b625] Bzip2_jll v1.0.9+0
[83423d85] Cairo_jll v1.18.7+0
[ee1fde0b] Dbus_jll v1.16.2+0
[2702e6a9] EpollShim_jll v0.0.20230411+1
[2e619515] Expat_jll v2.8.4+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
[f5851436] FFTW_jll v3.3.12+0
[a3f928ae] Fontconfig_jll v2.17.1+0
[d7e528f0] FreeType2_jll v2.14.3+1
[559328eb] FriBidi_jll v1.0.17+0
[0656b61e] GLFW_jll v3.5.1+0
[d2c73de3] GR_jll v0.73.27+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
[61579ee1] Ghostscript_jll v9.55.1+0
[7746bdde] Glib_jll v2.88.3+0
[3b182d85] Graphite2_jll v1.3.16+0
[2e76f6c2] HarfBuzz_jll v100.14004.0+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
[aacddb02] JpegTurbo_jll v3.2.0+1
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.2.0+0
[1d63c593] LLVMOpenMP_jll v23.1.1+0
[aae0fff6] LSODA_jll v0.1.2+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.42.0+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.0+0
[856f044c] MKL_jll v2025.2.0+0
[c771fb93] ODEInterface_jll v0.0.2+0
[e7412a2a] Ogg_jll v1.3.6+0
[656ef2d0] OpenBLAS32_jll v0.3.34+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.58.2+0
[30392449] Pixman_jll v0.46.4+0
[c0090381] Qt6Base_jll v6.10.2+2
[629bc702] Qt6Declarative_jll v6.10.2+2
[ce943373] Qt6ShaderTools_jll v6.10.2+1
[6de9746b] Qt6Svg_jll v6.10.2+0
[e99dba38] Qt6Wayland_jll v6.10.2+1
[f50d1b31] Rmath_jll v0.5.2+0
[ca45d3f4] SuiteSparse32_jll v7.12.1+1
[fb77eaff] Sundials_jll v7.5.0+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
[ffd25f8a] XZ_jll v5.8.4+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.13+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[935fb764] Xorg_libXcursor_jll v1.2.4+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.8+0
[d091e8ba] Xorg_libXfixes_jll v6.0.2+0
[a51aa0fd] Xorg_libXi_jll v1.8.4+0
[d1454406] Xorg_libXinerama_jll v1.1.7+0
[ec84b674] Xorg_libXrandr_jll v1.5.6+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+0
[a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[cc61e674] Xorg_libxkbfile_jll v1.2.0+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
[12413925] Xorg_xcb_util_image_jll v0.4.1+0
[2def613f] Xorg_xcb_util_jll v0.4.1+0
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
[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
[8f1865be] ZeroMQ_jll v4.3.6+0
[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
[a9144af2] libsodium_jll v1.0.21+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 v1.0.0
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.13.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.3.0
[44cfe95a] Pkg v1.13.0
[de0858da] Printf v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v1.0.0
[9e88b42a] Serialization v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.13.0
[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 v1.5.5+2
[deac9b47] LibCURL_jll v8.18.0+1
[e37daf67] LibGit2_jll v1.9.1+0
[29816b5a] LibSSH2_jll v1.11.103+0
[14a3606d] MozillaCACerts_jll v2026.8.13
[4536629a] OpenBLAS_jll v0.3.30+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.6+0
[efcefdf7] PCRE2_jll v10.46.0+0
[bea87d4a] SuiteSparse_jll v7.10.1+0
[83775a58] Zlib_jll v1.3.1+2
[3161d3a3] Zstd_jll v1.5.7+1
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.67.1+0
[3f19e933] p7zip_jll v17.8.2+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`