Quadruple Boson Energy Conservation

In this notebook we will study the energy conservation properties of several high-order methods for a system with the following Hamiltonian:

\[\mathcal{H}\left(q_0,q_2,p_0,p_2\right) = \frac{A}{2} \left(p_0^2 + p_2^2 + q_0^2 + q_2^2\right) + \frac{B}{\sqrt{2}} q_0 \left(3q_2^2 - q_0^2\right) + \frac{D}{4} \left(q_0^2+q_2^2\right)^2\]

This Hamiltonian resembles the Hénon-Heiles one, but it has an additional fourth order term. The aim of this benchmark is to see what happens with the energy error when highly accurate solutions are needed and how the results compare with the Hénon-Heiles case.

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

T(p) = A / 2 * norm(p)^2
V(q) = A / 2 * (q[1]^2 + q[2]^2) + B / sqrt(2) * q[1] * (3 * q[2]^2 - q[1]^2) + D / 4 * (q[1]^2 + q[2]^2)^2
H(p, q, params) = T(p) + V(q)

const A, B, D = 1., 0.55, 0.4

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

function iip_dp(dp, p, q, params, t)
    dp[1] = -A * q[1] - 3 * B / sqrt(2) * (q[2]^2 - q[1]^2) - D * q[1] * (q[1]^2 + q[2]^2)
    dp[2] = -q[2] * (A + 3 * sqrt(2) * B * q[1] + D * (q[1]^2 + q[2]^2))
end

const iip_q0 = [4.919080920016389, 2.836942666663649]
const iip_p0 = [0., 0.]
const iip_u0 = vcat(iip_p0,iip_q0)

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

function oop_dp(p, q, params, t)
    dp1 = -A * q[1] - 3 * B / sqrt(2) * (q[2]^2 - q[1]^2) - D * q[1] * (q[1]^2 + q[2]^2)
    dp2 = -q[2] * (A + 3 * sqrt(2) * B * q[1] + D * (q[1]^2 + q[2]^2))
    @SVector [dp1, dp2]
end

const oop_q0 = @SVector [4.919080920016389, 2.836942666663649]
const oop_p0 = @SVector [0., 0.]
const oop_u0 = vcat(oop_p0,oop_q0)

function hamilton(z, params, t)
    SVector(
    -A * z[3] - 3 * B / sqrt(2) * (z[4]^2 - z[3]^2) - D * z[3] * (z[3]^2 + z[4]^2),
    -z[4] * (A + 3 * sqrt(2) * B * z[3] + D * (z[3]^2 + z[4]^2)),
    z[1],
    z[2]
    )
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 E = H(iip_p0, iip_q0, nothing)
const cb = ManifoldProjection(g, nlopts=Dict(:ftol=>1e-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)
272.019764 seconds (215.35 M allocations: 11.276 GiB, 1.44% gc time, 99.97%
 compilation time)
  4.354348 seconds (15.45 M allocations: 776.012 MiB, 2.86% gc time, 99.82%
 compilation time)
  3.498601 seconds (8.31 M allocations: 436.694 MiB, 1.74% gc time, 99.69% 
compilation time)
172.771503 seconds (19.61 M allocations: 790.264 MiB, 0.16% gc time, 99.98%
 compilation time)
  5.033916 seconds (8.93 M allocations: 483.348 MiB, 1.32% gc time, 99.93% 
compilation time)
  9.770452 seconds (24.34 M allocations: 2.442 GiB, 5.00% gc time, 97.83% c
ompilation time)
Vern9 + ManifoldProjection max energy error:	1.5631940186722204e-13	in	9731
	steps.
KahanLi8 max energy error:			5.5706550483591855e-12	in	10001	steps.
SofSpa10 max energy error:			3.836930773104541e-12	in	10001	steps.
Vern9 max energy error:				1.4779288903810084e-12	in	4865	steps.
DPRKN12 max energy error:			6.252776074688882e-13	in	2195	steps.
TaylorMethod max energy error:			2.8421709430404007e-13	in	509	steps.

compare(tmax=1e3)
0.895569 seconds (5.34 M allocations: 364.811 MiB, 15.90% gc time)
  0.080908 seconds (700.10 k allocations: 61.899 MiB)
  0.115533 seconds (700.10 k allocations: 61.901 MiB)
  0.290910 seconds (1.70 M allocations: 106.026 MiB)
  0.032407 seconds (415.82 k allocations: 18.253 MiB)
  1.589075 seconds (8.15 M allocations: 2.035 GiB, 10.69% gc time)
Vern9 + ManifoldProjection max energy error:	2.2737367544323206e-13	in	9726
5	steps.
KahanLi8 max energy error:			1.0530243343964685e-11	in	100002	steps.
SofSpa10 max energy error:			1.5077716852829326e-11	in	100002	steps.
Vern9 max energy error:				7.361222742474638e-12	in	48626	steps.
DPRKN12 max energy error:			3.197442310920451e-12	in	21875	steps.
TaylorMethod max energy error:			1.3358203432289883e-12	in	5082	steps.

compare(tmax=1e4)
9.666883 seconds (53.41 M allocations: 3.530 GiB, 20.24% gc time)
  0.829493 seconds (7.00 M allocations: 579.841 MiB)
  1.163412 seconds (7.00 M allocations: 579.844 MiB)
  2.918431 seconds (17.02 M allocations: 1.020 GiB)
  2.856208 seconds (4.16 M allocations: 178.283 MiB, 88.35% gc time)
 14.974092 seconds (81.51 M allocations: 20.348 GiB, 27.36% gc time)
Vern9 + ManifoldProjection max energy error:	2.2737367544323206e-13	in	9723
87	steps.
KahanLi8 max energy error:			4.3968384488835e-11	in	1000001	steps.
SofSpa10 max energy error:			6.492939519375795e-11	in	1000001	steps.
Vern9 max energy error:				6.2399863054452e-11	in	486191	steps.
DPRKN12 max energy error:			1.496403001510771e-11	in	218676	steps.
TaylorMethod max energy error:			5.4001247917767614e-12	in	50814	steps.

compare(tmax=2e4)
21.934919 seconds (106.82 M allocations: 7.058 GiB, 29.20% gc time)
  1.681581 seconds (14.00 M allocations: 1.171 GiB)
 10.982748 seconds (14.00 M allocations: 1.171 GiB, 78.72% gc time)
  9.205334 seconds (34.03 M allocations: 2.037 GiB, 36.62% gc time)
  4.934079 seconds (8.31 M allocations: 354.213 MiB, 86.79% gc time)
 35.555383 seconds (163.01 M allocations: 40.697 GiB, 38.82% gc time)
Vern9 + ManifoldProjection max energy error:	2.2737367544323206e-13	in	1944
763	steps.
KahanLi8 max energy error:			1.0363976343796821e-10	in	2000002	steps.
SofSpa10 max energy error:			9.750067420100095e-11	in	2000002	steps.
Vern9 max energy error:				1.2180123576399637e-10	in	972387	steps.
DPRKN12 max energy error:			4.125411123823142e-11	in	437335	steps.
TaylorMethod max energy error:			1.4424017535930034e-11	in	101627	steps.

As we can see from the above plots, we can achieve a very low energy error for long time simulation by manifold projection and with very high order Taylor methods. In comparison with the Hénon-Heiles system we see that as the Hamiltonian got more complex, the energy error for the other integration methods increased significantly.

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:
  0.823618 seconds (5.34 M allocations: 364.811 MiB, 7.59% gc time)
  0.076931 seconds (700.10 k allocations: 61.899 MiB)
  0.115324 seconds (700.10 k allocations: 61.901 MiB)
  0.292947 seconds (1.70 M allocations: 106.026 MiB)
  0.032784 seconds (415.82 k allocations: 18.253 MiB)
  1.636513 seconds (8.15 M allocations: 2.035 GiB, 11.44% gc time)
Vern9 + ManifoldProjection max energy error:	2.2737367544323206e-13	in	9726
5	steps.
KahanLi8 max energy error:			1.0530243343964685e-11	in	100002	steps.
SofSpa10 max energy error:			1.5077716852829326e-11	in	100002	steps.
Vern9 max energy error:				7.361222742474638e-12	in	48626	steps.
DPRKN12 max energy error:			3.197442310920451e-12	in	21875	steps.
TaylorMethod max energy error:			1.3358203432289883e-12	in	5082	steps.

Out of place versions:
  1.904722 seconds (5.12 M allocations: 299.049 MiB, 3.46% gc time, 95.78% 
compilation time)
  1.176697 seconds (2.20 M allocations: 136.747 MiB, 93.17% compilation tim
e)
  5.587944 seconds (6.69 M allocations: 319.367 MiB, 0.73% gc time, 99.28% 
compilation time)
  1.835557 seconds (2.66 M allocations: 136.064 MiB, 99.03% compilation tim
e)
KahanLi8 max energy error:			1.0530243343964685e-11	in	100002	steps.
SofSpa10 max energy error:			1.5077716852829326e-11	in	100002	steps.
Vern9 max energy error:				9.450218385609332e-12	in	48622	steps.
DPRKN12 max energy error:			3.268496584496461e-12	in	21872	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:
  0.007709 seconds (70.09 k allocations: 5.806 MiB)
  0.010265 seconds (70.09 k allocations: 5.808 MiB)
  0.026899 seconds (170.46 k allocations: 10.880 MiB)
  0.003172 seconds (41.89 k allocations: 1.907 MiB)
KahanLi8 max energy error:			5.5706550483591855e-12	in	10001	steps.
SofSpa10 max energy error:			3.836930773104541e-12	in	10001	steps.
Vern9 max energy error:				1.4779288903810084e-12	in	4865	steps.
DPRKN12 max energy error:			6.252776074688882e-13	in	2195	steps.

Out of place versions:
  0.004591 seconds (10.05 k allocations: 1.682 MiB)
  0.008068 seconds (10.05 k allocations: 1.683 MiB)
  0.003495 seconds (73.05 k allocations: 3.810 MiB)
  0.001830 seconds (33.00 k allocations: 1.060 MiB)
KahanLi8 max energy error:			5.5706550483591855e-12	in	10001	steps.
SofSpa10 max energy error:			3.836930773104541e-12	in	10001	steps.
Vern9 max energy error:				5.542233338928781e-13	in	4864	steps.
DPRKN12 max energy error:			4.689582056016661e-13	in	2194	steps.

in_vs_out(tmax=1e3)
In place versions:
  0.069657 seconds (700.10 k allocations: 61.899 MiB)
  0.103680 seconds (700.10 k allocations: 61.901 MiB)
  0.292163 seconds (1.70 M allocations: 106.026 MiB)
  0.033341 seconds (415.82 k allocations: 18.253 MiB)
KahanLi8 max energy error:			1.0530243343964685e-11	in	100002	steps.
SofSpa10 max energy error:			1.5077716852829326e-11	in	100002	steps.
Vern9 max energy error:				7.361222742474638e-12	in	48626	steps.
DPRKN12 max energy error:			3.197442310920451e-12	in	21875	steps.

Out of place versions:
  0.048165 seconds (100.05 k allocations: 21.886 MiB)
  0.080662 seconds (100.05 k allocations: 21.887 MiB)
  0.037188 seconds (729.43 k allocations: 34.346 MiB)
  0.016776 seconds (328.18 k allocations: 9.662 MiB)
KahanLi8 max energy error:			1.0530243343964685e-11	in	100002	steps.
SofSpa10 max energy error:			1.5077716852829326e-11	in	100002	steps.
Vern9 max energy error:				9.450218385609332e-12	in	48622	steps.
DPRKN12 max energy error:			3.268496584496461e-12	in	21872	steps.

in_vs_out(tmax=1e4)
In place versions:
  1.213027 seconds (7.00 M allocations: 579.841 MiB, 33.44% gc time)
  1.428774 seconds (7.00 M allocations: 579.844 MiB, 19.29% gc time)
  3.309333 seconds (17.02 M allocations: 1.020 GiB, 11.76% gc time)
  0.331072 seconds (4.16 M allocations: 178.283 MiB)
KahanLi8 max energy error:			4.3968384488835e-11	in	1000001	steps.
SofSpa10 max energy error:			6.492939519375795e-11	in	1000001	steps.
Vern9 max energy error:				6.2399863054452e-11	in	486191	steps.
DPRKN12 max energy error:			1.496403001510771e-11	in	218676	steps.

Out of place versions:
  0.514200 seconds (1.00 M allocations: 167.850 MiB, 5.97% gc time)
  0.800543 seconds (1.00 M allocations: 167.851 MiB)
  0.392583 seconds (7.29 M allocations: 319.629 MiB)
  0.176962 seconds (3.28 M allocations: 90.035 MiB)
KahanLi8 max energy error:			4.3968384488835e-11	in	1000001	steps.
SofSpa10 max energy error:			6.492939519375795e-11	in	1000001	steps.
Vern9 max energy error:				5.697131655324483e-11	in	486200	steps.
DPRKN12 max energy error:			2.609112925711088e-11	in	218667	steps.

in_vs_out(tmax=2e4)
In place versions:
  2.116293 seconds (14.00 M allocations: 1.171 GiB, 21.81% gc time)
  2.312367 seconds (14.00 M allocations: 1.171 GiB)
  6.782142 seconds (34.03 M allocations: 2.037 GiB, 13.95% gc time)
  3.102480 seconds (8.31 M allocations: 354.213 MiB, 78.76% gc time)
KahanLi8 max energy error:			1.0363976343796821e-10	in	2000002	steps.
SofSpa10 max energy error:			9.750067420100095e-11	in	2000002	steps.
Vern9 max energy error:				1.2180123576399637e-10	in	972387	steps.
DPRKN12 max energy error:			4.125411123823142e-11	in	437335	steps.

Out of place versions:
  1.069182 seconds (2.00 M allocations: 395.141 MiB, 7.19% gc time)
  1.617748 seconds (2.00 M allocations: 395.142 MiB)
  0.976961 seconds (14.59 M allocations: 636.323 MiB, 20.03% gc time)
  0.355404 seconds (6.56 M allocations: 177.099 MiB)
KahanLi8 max energy error:			1.0363976343796821e-10	in	2000002	steps.
SofSpa10 max energy error:			9.750067420100095e-11	in	2000002	steps.
Vern9 max energy error:				1.2643397440115223e-10	in	972386	steps.
DPRKN12 max energy error:			4.7620574150641914e-11	in	437330	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. In comparison with the Henon-Heiles case, we see that the symplectic methods are more competitive with DPRKN12.

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","Quadrupole_boson_Hamiltonian_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
  [e2ed5e7c] Bijections v0.1.4
  [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
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