Hénon-Heiles Energy Conservation

In this notebook we will study the energy conservation properties of several high-order methods for the Hénon-Heiles system. We will se how the energy error behaves at very thight tolerances and how different techniques, such as using symplectic solvers or manifold projections, benchmark against each other. The Hamiltonian for this system is given by:

\[\mathcal{H}=\frac{1}{2}(p_1^2 + p_2^2) + \frac{1}{2}\left(q_1^2 + q_2^2 + 2q_1^2 q_2 - \frac{2}{3}q_2^3\right)\]

We will also compare the in place apporach with the out of place approach by using Arrays (for the in place version) and StaticArrays (for out of place versions). In order to separate these two, we will use iip for the in-place names and oop for out of place ones.

using OrdinaryDiffEq, Plots, DiffEqCallbacks
using SciMLBenchmarks
using TaylorIntegration, LinearAlgebra, StaticArrays
gr(fmt=:png)
default(fmt=:png)

T(p) = 1//2 * norm(p)^2
V(q) = 1//2 * (q[1]^2 + q[2]^2 + 2q[1]^2 * q[2]- 2//3 * q[2]^3)
H(p,q, params) = T(p) + V(q)

function iip_dq(dq,p,q,params,t)
    dq[1] = p[1]
    dq[2] = p[2]
end

function iip_dp(dp,p,q,params,t)
    dp[1] = -q[1] * (1 + 2q[2])
    dp[2] = -q[2] - (q[1]^2 - q[2]^2)
end

const iip_q0 = [0.1, 0.]
const iip_p0 = [0., 0.5]

function oop_dq(p, q, params, t)
    p
end

function oop_dp(p, q, params, t)
    dp1 = -q[1] * (1 + 2q[2])
    dp2 = -q[2] - (q[1]^2 - q[2]^2)
    @SVector [dp1, dp2]
end

const oop_q0 = @SVector [0.1, 0.]
const oop_p0 = @SVector [0., 0.5]

function hamilton(du,u,p,t)
    dq, q = @views u[3:4], du[3:4]
    dp, p = @views u[1:2], du[1:2]

    dp[1] = -q[1] * (1 + 2q[2])
    dp[2] = -q[2] - (q[1]^2 - q[2]^2)
    dq .= p

    return nothing
end

function g(resid, u, p)
    resid[1] = H([u[1],u[2]], [u[3],u[4]], nothing) - E
    resid[2:4] .= 0
end

const cb = ManifoldProjection(g, nlopts=Dict(:ftol=>1e-13))

const E = H(iip_p0, iip_q0, nothing)
0.13

For the comparison we will use the following function

energy_err(sol) = map(i->H([sol[1,i], sol[2,i]], [sol[3,i], sol[4,i]], nothing)-E, 1:length(sol.u))
abs_energy_err(sol) = [abs.(H([sol[1,j], sol[2,j]], [sol[3,j], sol[4,j]], nothing) - E) for j=1:length(sol.u)]

function compare(mode=:inplace, all=true, plt=nothing; tmax=1e2)
    if mode == :inplace
        prob = DynamicalODEProblem(iip_dp, iip_dq, iip_p0, iip_q0, (0., tmax))
    else
        prob = DynamicalODEProblem(oop_dp, oop_dq, oop_p0, oop_q0, (0., tmax))
    end
    prob_linear = ODEProblem(hamilton, vcat(iip_p0, iip_q0), (0., tmax))

    GC.gc()
    (mode == :inplace && all) && @time sol1 = solve(prob, Vern9(), callback=cb, abstol=1e-14, reltol=1e-14)
    GC.gc()
    @time sol2 = solve(prob, KahanLi8(), dt=1e-2, maxiters=1e10)
    GC.gc()
    @time sol3 = solve(prob, SofSpa10(), dt=1e-2, maxiters=1e8)
    GC.gc()
    @time sol4 = solve(prob, Vern9(), abstol=1e-14, reltol=1e-14)
    GC.gc()
    @time sol5 = solve(prob, DPRKN12(), abstol=1e-14, reltol=1e-14)
    GC.gc()
    (mode == :inplace && all) && @time sol6 = solve(prob_linear, TaylorMethod(50), abstol=1e-20)

    (mode == :inplace && all) && println("Vern9 + ManifoldProjection max energy error:\t"*
        "$(maximum(abs_energy_err(sol1)))\tin\t$(length(sol1.u))\tsteps.")
    println("KahanLi8 max energy error:\t\t\t$(maximum(abs_energy_err(sol2)))\tin\t$(length(sol2.u))\tsteps.")
    println("SofSpa10 max energy error:\t\t\t$(maximum(abs_energy_err(sol3)))\tin\t$(length(sol3.u))\tsteps.")
    println("Vern9 max energy error:\t\t\t\t$(maximum(abs_energy_err(sol4)))\tin\t$(length(sol4.u))\tsteps.")
    println("DPRKN12 max energy error:\t\t\t$(maximum(abs_energy_err(sol5)))\tin\t$(length(sol5.u))\tsteps.")
    (mode == :inplace && all) && println("TaylorMethod max energy error:\t\t\t$(maximum(abs_energy_err(sol6)))"*
        "\tin\t$(length(sol6.u))\tsteps.")

    if plt === nothing
        plt = plot(xlabel="t", ylabel="Energy error")
    end
    (mode == :inplace && all) && plot!(sol1.t, energy_err(sol1), label="Vern9 + ManifoldProjection")
    plot!(sol2.t, energy_err(sol2), label="KahanLi8", ls=mode==:inplace ? :solid : :dash)
    plot!(sol3.t, energy_err(sol3), label="SofSpa10", ls=mode==:inplace ? :solid : :dash)
    plot!(sol4.t, energy_err(sol4), label="Vern9", ls=mode==:inplace ? :solid : :dash)
    plot!(sol5.t, energy_err(sol5), label="DPRKN12", ls=mode==:inplace ? :solid : :dash)
    (mode == :inplace && all) && plot!(sol6.t, energy_err(sol6), label="TaylorMethod")

    return plt
end
compare (generic function with 4 methods)

The mode argument choses between the in place approach and the out of place one. The all parameter is used to compare only the integrators that support both the in place and the out of place versions (we reffer here only to the 6 high order methods chosen bellow). The plt argument can be used to overlay the results over a previous plot and the tmax keyword determines the simulation time.

Note:

  1. The Vern9 method is used with ODEProblem because of performance issues with ArrayPartition indexing which manifest for DynamicalODEProblem.
  2. The NLsolve call used by ManifoldProjection was modified to use ftol=1e-13 in order to obtain a very low energy error.

Here are the results of the comparisons between the in place methods:

compare(tmax=1e2)
271.699670 seconds (212.71 M allocations: 10.770 GiB, 1.44% gc time, 99.99%
 compilation time)
  4.036090 seconds (14.93 M allocations: 744.111 MiB, 3.35% gc time, 99.82%
 compilation time)
  3.229579 seconds (7.91 M allocations: 411.130 MiB, 2.04% gc time, 99.68% 
compilation time)
175.349204 seconds (19.33 M allocations: 772.018 MiB, 0.15% gc time, 100.00
% compilation time)
  5.191661 seconds (8.42 M allocations: 452.258 MiB, 0.84% gc time, 99.97% 
compilation time)
  2.498380 seconds (5.79 M allocations: 398.278 MiB, 2.19% gc time, 99.97% 
compilation time)
Vern9 + ManifoldProjection max energy error:	3.0531133177191805e-16	in	1881
	steps.
KahanLi8 max energy error:			4.9404924595819466e-15	in	10001	steps.
SofSpa10 max energy error:			5.440092820663267e-15	in	10001	steps.
Vern9 max energy error:				3.0531133177191805e-16	in	941	steps.
DPRKN12 max energy error:			1.942890293094024e-16	in	385	steps.
TaylorMethod max energy error:			0.0	in	2	steps.

compare(tmax=1e3)
0.108232 seconds (1.02 M allocations: 69.054 MiB)
  0.076456 seconds (700.10 k allocations: 61.899 MiB)
  0.108398 seconds (700.10 k allocations: 61.901 MiB)
  0.053855 seconds (326.73 k allocations: 20.282 MiB)
  0.005685 seconds (72.14 k allocations: 3.121 MiB)
  0.000586 seconds (2.11 k allocations: 785.609 KiB)
Vern9 + ManifoldProjection max energy error:	4.274358644806853e-15	in	18659
	steps.
KahanLi8 max energy error:			1.815214645262131e-14	in	100002	steps.
SofSpa10 max energy error:			2.8033131371785203e-14	in	100002	steps.
Vern9 max energy error:				4.274358644806853e-15	in	9330	steps.
DPRKN12 max energy error:			8.604228440844963e-16	in	3787	steps.
TaylorMethod max energy error:			0.0	in	2	steps.

compare(tmax=1e4)
1.453077 seconds (10.16 M allocations: 685.649 MiB, 19.38% gc time)
  0.969164 seconds (7.00 M allocations: 579.841 MiB, 21.58% gc time)
  1.062469 seconds (7.00 M allocations: 579.844 MiB)
  0.532087 seconds (3.26 M allocations: 203.204 MiB)
  0.054207 seconds (718.64 k allocations: 32.470 MiB)
  0.000576 seconds (2.11 k allocations: 785.609 KiB)
Vern9 + ManifoldProjection max energy error:	4.593547764386585e-14	in	18642
9	steps.
KahanLi8 max energy error:			3.161360062620133e-14	in	1000001	steps.
SofSpa10 max energy error:			1.136590821460004e-13	in	1000001	steps.
Vern9 max energy error:				4.593547764386585e-14	in	93215	steps.
DPRKN12 max energy error:			8.465450562766819e-15	in	37813	steps.
TaylorMethod max energy error:			0.0	in	2	steps.

compare(tmax=5e4)
7.725003 seconds (51.14 M allocations: 3.347 GiB, 21.51% gc time, 1.61% c
ompilation time)
  6.333915 seconds (35.00 M allocations: 2.831 GiB, 39.27% gc time)
 10.674456 seconds (35.00 M allocations: 2.831 GiB, 50.43% gc time)
  2.671526 seconds (16.31 M allocations: 1001.925 MiB)
  4.839273 seconds (3.59 M allocations: 155.672 MiB, 94.27% gc time)
  0.000634 seconds (2.11 k allocations: 785.609 KiB)
Vern9 + ManifoldProjection max energy error:	1.0000333894311098e-13	in	9320
91	steps.
KahanLi8 max energy error:			1.2331802246023926e-13	in	5000001	steps.
SofSpa10 max energy error:			1.5035195310986182e-13	in	5000001	steps.
Vern9 max energy error:				2.2451485115482228e-13	in	466046	steps.
DPRKN12 max energy error:			3.5388358909926865e-14	in	189039	steps.
TaylorMethod max energy error:			0.0	in	2	steps.

We can see that as the simulation time increases, the energy error increases. For this particular example the energy error for all the methods is comparable. For relatively short simulation times, if a highly accurate solution is required, the symplectic method is not recommended as its energy error fluctuations are larger than for other methods. An other thing to notice is the fact that the two versions of Vern9 behave identically, as expected, untill the energy error set by ftol is reached.

We will now compare the in place with the out of place versions. In the plots bellow we will use a dashed line for the out of place versions.

function in_vs_out(;all=false, tmax=1e2)
    println("In place versions:")
    plt = compare(:inplace, all, tmax=tmax)
    println("\nOut of place versions:")
    plt = compare(:oop, false, plt; tmax=tmax)
end
in_vs_out (generic function with 1 method)

First, here is a summary of all the available methods for tmax = 1e3:

in_vs_out(all=true, tmax=1e3)
In place versions:
  0.108057 seconds (1.02 M allocations: 69.054 MiB)
  0.075184 seconds (700.10 k allocations: 61.899 MiB)
  0.108528 seconds (700.10 k allocations: 61.901 MiB)
  0.054281 seconds (326.73 k allocations: 20.282 MiB)
  0.005650 seconds (72.14 k allocations: 3.121 MiB)
  0.000588 seconds (2.11 k allocations: 785.609 KiB)
Vern9 + ManifoldProjection max energy error:	4.274358644806853e-15	in	18659
	steps.
KahanLi8 max energy error:			1.815214645262131e-14	in	100002	steps.
SofSpa10 max energy error:			2.8033131371785203e-14	in	100002	steps.
Vern9 max energy error:				4.274358644806853e-15	in	9330	steps.
DPRKN12 max energy error:			8.604228440844963e-16	in	3787	steps.
TaylorMethod max energy error:			0.0	in	2	steps.

Out of place versions:
  1.922603 seconds (5.20 M allocations: 303.006 MiB, 3.82% gc time, 98.59% 
compilation time)
  1.107676 seconds (2.23 M allocations: 137.174 MiB, 2.30% gc time, 94.33% 
compilation time)
  5.604480 seconds (6.14 M allocations: 292.238 MiB, 0.63% gc time, 99.89% 
compilation time)
  1.817120 seconds (2.39 M allocations: 127.359 MiB, 99.86% compilation tim
e)
KahanLi8 max energy error:			1.815214645262131e-14	in	100002	steps.
SofSpa10 max energy error:			2.8033131371785203e-14	in	100002	steps.
Vern9 max energy error:				4.496403249731884e-15	in	9330	steps.
DPRKN12 max energy error:			4.440892098500626e-16	in	3787	steps.

Now we will compare the in place and the out of place versions, but only for the integrators that are compatible with StaticArrays

in_vs_out(tmax=1e2)
In place versions:
  0.007093 seconds (70.09 k allocations: 5.806 MiB)
  0.009447 seconds (70.09 k allocations: 5.808 MiB)
  0.005102 seconds (33.11 k allocations: 2.100 MiB)
  0.000674 seconds (7.50 k allocations: 352.312 KiB)
KahanLi8 max energy error:			4.9404924595819466e-15	in	10001	steps.
SofSpa10 max energy error:			5.440092820663267e-15	in	10001	steps.
Vern9 max energy error:				3.0531133177191805e-16	in	941	steps.
DPRKN12 max energy error:			1.942890293094024e-16	in	385	steps.

Out of place versions:
  0.002567 seconds (10.05 k allocations: 1.682 MiB)
  0.003766 seconds (10.05 k allocations: 1.683 MiB)
  0.000658 seconds (14.20 k allocations: 716.672 KiB)
  0.000330 seconds (5.86 k allocations: 184.438 KiB)
KahanLi8 max energy error:			4.9404924595819466e-15	in	10001	steps.
SofSpa10 max energy error:			5.440092820663267e-15	in	10001	steps.
Vern9 max energy error:				3.3306690738754696e-16	in	941	steps.
DPRKN12 max energy error:			2.220446049250313e-16	in	385	steps.

in_vs_out(tmax=1e3)
In place versions:
  0.066575 seconds (700.10 k allocations: 61.899 MiB)
  0.108494 seconds (700.10 k allocations: 61.901 MiB)
  0.055988 seconds (326.73 k allocations: 20.282 MiB)
  0.005822 seconds (72.14 k allocations: 3.121 MiB)
KahanLi8 max energy error:			1.815214645262131e-14	in	100002	steps.
SofSpa10 max energy error:			2.8033131371785203e-14	in	100002	steps.
Vern9 max energy error:				4.274358644806853e-15	in	9330	steps.
DPRKN12 max energy error:			8.604228440844963e-16	in	3787	steps.

Out of place versions:
  0.027341 seconds (100.05 k allocations: 21.886 MiB)
  0.036937 seconds (100.05 k allocations: 21.887 MiB)
  0.006288 seconds (140.04 k allocations: 6.467 MiB)
  0.002424 seconds (56.89 k allocations: 1.595 MiB)
KahanLi8 max energy error:			1.815214645262131e-14	in	100002	steps.
SofSpa10 max energy error:			2.8033131371785203e-14	in	100002	steps.
Vern9 max energy error:				4.496403249731884e-15	in	9330	steps.
DPRKN12 max energy error:			4.440892098500626e-16	in	3787	steps.

in_vs_out(tmax=1e4)
In place versions:
  1.094424 seconds (7.00 M allocations: 579.841 MiB, 29.10% gc time)
  1.358227 seconds (7.00 M allocations: 579.844 MiB, 21.46% gc time)
  0.539791 seconds (3.26 M allocations: 203.204 MiB)
  0.054564 seconds (718.64 k allocations: 32.470 MiB)
KahanLi8 max energy error:			3.161360062620133e-14	in	1000001	steps.
SofSpa10 max energy error:			1.136590821460004e-13	in	1000001	steps.
Vern9 max energy error:				4.593547764386585e-14	in	93215	steps.
DPRKN12 max energy error:			8.465450562766819e-15	in	37813	steps.

Out of place versions:
  0.299182 seconds (1.00 M allocations: 167.850 MiB, 7.17% gc time)
  0.361995 seconds (1.00 M allocations: 167.851 MiB)
  0.066102 seconds (1.40 M allocations: 65.297 MiB)
  0.022775 seconds (567.29 k allocations: 18.100 MiB)
KahanLi8 max energy error:			3.161360062620133e-14	in	1000001	steps.
SofSpa10 max energy error:			1.136590821460004e-13	in	1000001	steps.
Vern9 max energy error:				4.4797499043625066e-14	in	93215	steps.
DPRKN12 max energy error:			2.6922908347160046e-15	in	37813	steps.

in_vs_out(tmax=5e4)
In place versions:
  5.850563 seconds (35.00 M allocations: 2.831 GiB, 33.95% gc time)
  9.546422 seconds (35.00 M allocations: 2.831 GiB, 44.57% gc time)
  2.665904 seconds (16.31 M allocations: 1001.925 MiB)
  3.930075 seconds (3.59 M allocations: 155.672 MiB, 92.98% gc time)
KahanLi8 max energy error:			1.2331802246023926e-13	in	5000001	steps.
SofSpa10 max energy error:			1.5035195310986182e-13	in	5000001	steps.
Vern9 max energy error:				2.2451485115482228e-13	in	466046	steps.
DPRKN12 max energy error:			3.5388358909926865e-14	in	189039	steps.

Out of place versions:
  1.559227 seconds (5.00 M allocations: 839.237 MiB, 12.25% gc time)
  2.026374 seconds (5.00 M allocations: 839.238 MiB, 10.26% gc time)
  0.322292 seconds (6.99 M allocations: 307.636 MiB)
  0.120090 seconds (2.84 M allocations: 80.089 MiB)
KahanLi8 max energy error:			1.2331802246023926e-13	in	5000001	steps.
SofSpa10 max energy error:			1.5035195310986182e-13	in	5000001	steps.
Vern9 max energy error:				2.246258734572848e-13	in	466047	steps.
DPRKN12 max energy error:			3.4361402612148595e-14	in	189040	steps.

As we see from the above comparisons, the StaticArray versions are significantly faster and use less memory. The speedup provided for the out of place version is more proeminent at larger values for tmax. We can see again that if the simulation time is increased, the energy error of the symplectic methods is less noticeable compared to the rest of the methods.

The benchmarks were performed on a machine with

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/DynamicalODE","Henon-Heiles_energy_conservation_benchmark.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:
  JULIA_CPU_THREADS = 128
  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/DynamicalODE/Project.toml`
  [459566f4] DiffEqCallbacks v2.23.1
  [055956cb] DiffEqPhysics v3.9.0
  [b305315f] Elliptic v1.0.1
  [1dea7af3] OrdinaryDiffEq v6.19.1
  [65888b18] ParameterizedFunctions v5.13.2
  [91a5bcdd] Plots v1.31.4
  [d330b81b] PyPlot v2.10.0
  [31c91b34] SciMLBenchmarks v0.1.0
  [90137ffa] StaticArrays v1.5.2
  [92b13dbe] TaylorIntegration v0.8.11
  [37e2e46d] LinearAlgebra
  [de0858da] Printf
  [10745b16] Statistics

And the full manifest:

      Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DynamicalODE/Manifest.toml`
  [c3fe647b] AbstractAlgebra v0.27.0
  [1520ce14] AbstractTrees v0.4.2
  [79e6a3ab] Adapt v3.3.3
  [dce04be8] ArgCheck v2.3.0
  [ec485272] ArnoldiMethod v0.2.0
  [4fba245c] ArrayInterface v6.0.21
  [30b0a656] ArrayInterfaceCore v0.1.15
  [6ba088a2] ArrayInterfaceGPUArrays v0.2.1
  [015c0d05] ArrayInterfaceOffsetArrays v0.1.6
  [b0d46f97] ArrayInterfaceStaticArrays v0.1.4
  [dd5226c6] ArrayInterfaceStaticArraysCore v0.1.0
  [15f4f7f2] AutoHashEquals v0.2.0
  [198e06fe] BangBang v0.3.36
  [9718e550] Baselet v0.1.1
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  [62783981] BitTwiddlingConvenienceFunctions v0.1.4
  [2a0fbf3d] CPUSummary v0.1.25
  [00ebfdb7] CSTParser v3.3.6
  [49dc2e85] Calculus v0.5.1
  [d360d2e6] ChainRulesCore v1.15.3
  [9e997f8a] ChangesOfVariables v0.1.4
  [fb6a15b2] CloseOpenIntervals v0.1.10
  [944b1d66] CodecZlib v0.7.0
  [35d6a980] ColorSchemes v3.19.0
  [3da002f7] ColorTypes v0.11.4
  [c3611d14] ColorVectorSpace v0.9.9
  [5ae59095] Colors v0.12.8
  [861a8166] Combinatorics v1.0.2
  [a80b9123] CommonMark v0.8.6
  [38540f10] CommonSolve v0.2.1
  [bbf7d656] CommonSubexpressions v0.3.0
  [34da2185] Compat v3.45.0
  [b152e2b5] CompositeTypes v0.1.2
  [a33af91c] CompositionsBase v0.1.1
  [8f4d0f93] Conda v1.7.0
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  [d38c429a] Contour v0.6.2
  [adafc99b] CpuId v0.3.1
  [a8cc5b0e] Crayons v4.1.1
  [9a962f9c] DataAPI v1.10.0
  [864edb3b] DataStructures v0.18.13
  [e2d170a0] DataValueInterfaces v1.0.0
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  [b429d917] DensityInterface v0.4.0
  [2b5f629d] DiffEqBase v6.94.4
  [459566f4] DiffEqCallbacks v2.23.1
  [055956cb] DiffEqPhysics v3.9.0
  [163ba53b] DiffResults v1.0.3
  [b552c78f] DiffRules v1.11.0
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