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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

function g_jacobian(J, u, p)
    J[1, 1] = u[1]
    J[1, 2] = u[2]
    J[1, 3] = u[3]
    J[1, 4] = u[4]
    J[2:4, :] .= 0
end

const E = H(iip_p0, iip_q0, nothing)
const resid_prototype = zeros(4)
const cb = ManifoldProjection(g, manifold_jacobian=g_jacobian, resid_prototype=resid_prototype, nlopts=Dict(:ftol=>1e-13))
SciMLBase.DiscreteCallback{Returns{Bool}, DiffEqCallbacks.ManifoldProjectio
n{DiffEqCallbacks.UntypedNonAutonomousFunction{typeof(Main.var"##WeaveSandB
ox#225".g)}, DiffEqCallbacks.UntypedNonAutonomousFunction{typeof(Main.var"#
#WeaveSandBox#225".g_jacobian)}, Nothing, Missing, Base.Pairs{Symbol, Any, 
Tuple{Symbol, Symbol}, @NamedTuple{resid_prototype::Vector{Float64}, nlopts
::Dict{Symbol, Float64}}}, Nothing}, typeof(DiffEqCallbacks.initialize_mani
fold_projection), typeof(SciMLBase.FINALIZE_DEFAULT), Nothing}(Returns{Bool
}(true), DiffEqCallbacks.ManifoldProjection{DiffEqCallbacks.UntypedNonAuton
omousFunction{typeof(Main.var"##WeaveSandBox#225".g)}, DiffEqCallbacks.Unty
pedNonAutonomousFunction{typeof(Main.var"##WeaveSandBox#225".g_jacobian)}, 
Nothing, Missing, Base.Pairs{Symbol, Any, Tuple{Symbol, Symbol}, @NamedTupl
e{resid_prototype::Vector{Float64}, nlopts::Dict{Symbol, Float64}}}, Nothin
g}(DiffEqCallbacks.UntypedNonAutonomousFunction{typeof(Main.var"##WeaveSand
Box#225".g)}(false, Main.var"##WeaveSandBox#225".g, nothing), DiffEqCallbac
ks.UntypedNonAutonomousFunction{typeof(Main.var"##WeaveSandBox#225".g_jacob
ian)}(false, Main.var"##WeaveSandBox#225".g_jacobian, nothing), nothing, no
thing, missing, Base.Pairs{Symbol, Any, Tuple{Symbol, Symbol}, @NamedTuple{
resid_prototype::Vector{Float64}, nlopts::Dict{Symbol, Float64}}}(:resid_pr
ototype => [0.0, 0.0, 0.0, 0.0], :nlopts => Dict(:ftol => 1.0e-13)), nothin
g), DiffEqCallbacks.initialize_manifold_projection, SciMLBase.FINALIZE_DEFA
ULT, Bool[0, 1], nothing)

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)
49.657725 seconds (133.54 M allocations: 7.676 GiB, 10.17% gc time, 99.99%
 compilation time)
  1.922268 seconds (3.78 M allocations: 240.115 MiB, 99.55% compilation tim
e)
  2.088850 seconds (2.86 M allocations: 183.952 MiB, 2.30% gc time, 99.44% 
compilation time)
 29.172501 seconds (92.14 M allocations: 5.519 GiB, 11.80% gc time, 99.94% 
compilation time)
  2.803455 seconds (2.83 M allocations: 185.234 MiB, 99.87% compilation tim
e)
  2.030506 seconds (3.54 M allocations: 389.936 MiB, 3.17% gc time, 89.92% 
compilation time)
Vern9 + ManifoldProjection max energy error:	119.99999999999999	in	5	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:				6.679101716144942e-13	in	4865	steps.
DPRKN12 max energy error:			8.242295734817162e-13	in	2195	steps.
TaylorMethod max energy error:			4.689582056016661e-13	in	509	steps.

compare(tmax=1e3)
0.005777 seconds (28.43 k allocations: 65.900 MiB)
  0.074329 seconds (700.11 k allocations: 61.899 MiB)
  0.118619 seconds (700.11 k allocations: 61.901 MiB)
  0.178979 seconds (1.80 M allocations: 108.223 MiB)
  0.034558 seconds (459.63 k allocations: 19.256 MiB)
  1.712775 seconds (7.91 M allocations: 1.972 GiB, 10.34% gc time)
Vern9 + ManifoldProjection max energy error:	2.9100430434015755e7	in	5	step
s.
KahanLi8 max energy error:			1.0530243343964685e-11	in	100002	steps.
SofSpa10 max energy error:			1.5077716852829326e-11	in	100002	steps.
Vern9 max energy error:				7.545963853772264e-12	in	48624	steps.
DPRKN12 max energy error:			5.343281372915953e-12	in	21878	steps.
TaylorMethod max energy error:			1.9326762412674725e-12	in	5082	steps.

compare(tmax=1e4)
0.000705 seconds (10.39 k allocations: 636.797 KiB)
  1.794599 seconds (7.00 M allocations: 579.841 MiB, 51.57% gc time)
  1.960321 seconds (7.00 M allocations: 579.844 MiB, 34.75% gc time)
  2.036138 seconds (17.99 M allocations: 1.042 GiB, 15.75% gc time)
  0.358183 seconds (4.59 M allocations: 188.285 MiB)
 14.511228 seconds (79.07 M allocations: 19.715 GiB, 16.13% gc time)
Vern9 + ManifoldProjection max energy error:	119.99999999999999	in	5	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:				6.17461637375527e-11	in	486193	steps.
DPRKN12 max energy error:			3.127809122815961e-11	in	218667	steps.
TaylorMethod max energy error:			7.702283255639486e-12	in	50814	steps.

compare(tmax=2e4)
0.007325 seconds (28.39 k allocations: 65.899 MiB, 16.68% gc time)
  4.023579 seconds (14.00 M allocations: 1.171 GiB, 56.08% gc time)
  4.417920 seconds (14.00 M allocations: 1.171 GiB, 42.35% gc time)
 12.514150 seconds (35.98 M allocations: 2.081 GiB, 72.29% gc time)
  0.692874 seconds (9.18 M allocations: 374.223 MiB)
 30.406647 seconds (158.13 M allocations: 39.431 GiB, 20.90% gc time)
Vern9 + ManifoldProjection max energy error:	116.88152906230225	in	5	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:				1.1236522823310224e-10	in	972365	steps.
DPRKN12 max energy error:			4.389733021525899e-11	in	437324	steps.
TaylorMethod max energy error:			1.3926637620897964e-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:
  0.000688 seconds (10.39 k allocations: 636.797 KiB)
  0.075219 seconds (700.11 k allocations: 61.899 MiB)
  0.128604 seconds (700.11 k allocations: 61.901 MiB)
  0.177671 seconds (1.80 M allocations: 108.223 MiB)
  0.034107 seconds (459.63 k allocations: 19.256 MiB)
  1.708187 seconds (7.91 M allocations: 1.972 GiB, 10.01% gc time)
Vern9 + ManifoldProjection max energy error:	119.99999999999999	in	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:				7.545963853772264e-12	in	48624	steps.
DPRKN12 max energy error:			5.343281372915953e-12	in	21878	steps.
TaylorMethod max energy error:			1.9326762412674725e-12	in	5082	steps.

Out of place versions:
  1.384153 seconds (2.81 M allocations: 205.485 MiB, 2.12% gc time, 98.02% 
compilation time)
  0.722316 seconds (1.22 M allocations: 94.111 MiB, 93.88% compilation time
)
  1.460822 seconds (2.97 M allocations: 180.747 MiB, 2.41% gc time, 98.12% 
compilation time)
  0.866041 seconds (1.18 M allocations: 79.972 MiB, 98.69% compilation time
)
KahanLi8 max energy error:			1.0530243343964685e-11	in	100002	steps.
SofSpa10 max energy error:			1.5077716852829326e-11	in	100002	steps.
Vern9 max energy error:				9.762857189343777e-12	in	48624	steps.
DPRKN12 max energy error:			1.5205614545266144e-12	in	21873	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:
  0.007365 seconds (70.09 k allocations: 5.806 MiB)
  0.011099 seconds (70.09 k allocations: 5.808 MiB)
  0.014873 seconds (180.19 k allocations: 11.078 MiB)
  0.003308 seconds (46.28 k allocations: 2.006 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:				6.679101716144942e-13	in	4865	steps.
DPRKN12 max energy error:			8.242295734817162e-13	in	2195	steps.

Out of place versions:
  0.003085 seconds (10.02 k allocations: 1.681 MiB)
  0.004567 seconds (10.02 k allocations: 1.682 MiB)
  0.002411 seconds (4.90 k allocations: 2.762 MiB)
  0.001752 seconds (2.23 k allocations: 604.359 KiB)
KahanLi8 max energy error:			5.5706550483591855e-12	in	10001	steps.
SofSpa10 max energy error:			3.836930773104541e-12	in	10001	steps.
Vern9 max energy error:				1.5063505998114124e-12	in	4868	steps.
DPRKN12 max energy error:			4.263256414560601e-13	in	2194	steps.

in_vs_out(tmax=1e3)
In place versions:
  0.072615 seconds (700.11 k allocations: 61.899 MiB)
  0.120818 seconds (700.11 k allocations: 61.901 MiB)
  0.175355 seconds (1.80 M allocations: 108.223 MiB)
  0.034195 seconds (459.63 k allocations: 19.256 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:				7.545963853772264e-12	in	48624	steps.
DPRKN12 max energy error:			5.343281372915953e-12	in	21878	steps.

Out of place versions:
  0.027935 seconds (100.03 k allocations: 21.886 MiB)
  0.044203 seconds (100.03 k allocations: 21.887 MiB)
  0.022229 seconds (48.67 k allocations: 23.950 MiB)
  0.010747 seconds (21.91 k allocations: 4.989 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:				9.762857189343777e-12	in	48624	steps.
DPRKN12 max energy error:			1.5205614545266144e-12	in	21873	steps.

in_vs_out(tmax=1e4)
In place versions:
  1.796457 seconds (7.00 M allocations: 579.841 MiB, 52.63% gc time)
  1.883028 seconds (7.00 M allocations: 579.844 MiB, 32.66% gc time)
  2.050417 seconds (17.99 M allocations: 1.042 GiB, 15.90% gc time)
  0.347885 seconds (4.59 M allocations: 188.285 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:				6.17461637375527e-11	in	486193	steps.
DPRKN12 max energy error:			3.127809122815961e-11	in	218667	steps.

Out of place versions:
  0.323146 seconds (1.00 M allocations: 167.849 MiB, 6.51% gc time)
  0.477622 seconds (1.00 M allocations: 167.850 MiB)
  0.267688 seconds (486.25 k allocations: 215.754 MiB)
  0.111218 seconds (218.71 k allocations: 43.322 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:				8.169820375769632e-11	in	486197	steps.
DPRKN12 max energy error:			1.993782916542841e-11	in	218666	steps.

in_vs_out(tmax=2e4)
In place versions:
  2.848420 seconds (14.00 M allocations: 1.171 GiB, 38.45% gc time)
  4.409374 seconds (14.00 M allocations: 1.171 GiB, 42.40% gc time)
  9.225664 seconds (35.98 M allocations: 2.081 GiB, 62.49% gc time)
  0.680846 seconds (9.18 M allocations: 374.223 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:				1.1236522823310224e-10	in	972365	steps.
DPRKN12 max energy error:			4.389733021525899e-11	in	437324	steps.

Out of place versions:
  0.767162 seconds (2.00 M allocations: 395.141 MiB, 14.57% gc time)
  1.199778 seconds (2.00 M allocations: 395.141 MiB, 16.02% gc time)
  0.543190 seconds (972.45 k allocations: 428.591 MiB)
  0.220902 seconds (437.38 k allocations: 83.674 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:				1.5153034382819897e-10	in	972393	steps.
DPRKN12 max energy error:			4.1310954657092225e-11	in	437331	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 1.10.7
Commit 4976d05258e (2024-11-26 15:57 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
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
  JULIA_CPU_THREADS = 128
  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 v4.2.2
  [055956cb] DiffEqPhysics v3.15.0
  [b305315f] Elliptic v1.0.1
  [1dea7af3] OrdinaryDiffEq v6.90.1
  [65888b18] ParameterizedFunctions v5.17.0
  [91a5bcdd] Plots v1.40.9
  [31c91b34] SciMLBenchmarks v0.1.3
  [90137ffa] StaticArrays v1.9.10
  [92b13dbe] TaylorIntegration v0.16.2
  [37e2e46d] LinearAlgebra
  [de0858da] Printf
  [10745b16] Statistics v1.10.0

And the full manifest:

Status `/cache/build/exclusive-amdci3-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DynamicalODE/Manifest.toml`
  [47edcb42] ADTypes v1.11.0
  [1520ce14] AbstractTrees v0.4.5
  [7d9f7c33] Accessors v0.1.39
  [79e6a3ab] Adapt v4.1.1
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  [d7ba0133] Git v1.3.1
  [c27321d9] Glob v1.3.1
  [86223c79] Graphs v1.12.0
  [42e2da0e] Grisu v1.0.2
  [cd3eb016] HTTP v1.10.15
  [eafb193a] Highlights v0.5.3
  [3e5b6fbb] HostCPUFeatures v0.1.17
  [34004b35] HypergeometricFunctions v0.3.25
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  [1019f520] JLFzf v0.1.9
  [692b3bcd] JLLWrappers v1.7.0
  [682c06a0] JSON v0.21.4
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  [9286f039] OrdinaryDiffEqExplicitRK v1.1.0
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  [b0944070] OrdinaryDiffEqLowStorageRK v1.2.1
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  [43230ef6] OrdinaryDiffEqRosenbrock v1.4.0
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  [358294b1] OrdinaryDiffEqStabilizedRK v1.1.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v1.1.0
  [b1df2697] OrdinaryDiffEqTsit5 v1.1.0
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  [ae029012] Requires v1.3.0
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  [727e6d20] SimpleNonlinearSolve v2.1.0
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  [41fe7b60] Unzip v0.2.0
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  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.12
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  [6e34b625] Bzip2_jll v1.0.8+3
  [83423d85] Cairo_jll v1.18.2+1
  [ee1fde0b] Dbus_jll v1.14.10+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
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⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
  [a3f928ae] Fontconfig_jll v2.15.0+0
  [d7e528f0] FreeType2_jll v2.13.3+1
  [559328eb] FriBidi_jll v1.0.16+0
  [0656b61e] GLFW_jll v3.4.0+2
  [d2c73de3] GR_jll v0.73.10+0
  [78b55507] Gettext_jll v0.21.0+0
  [f8c6e375] Git_jll v2.47.1+0
  [7746bdde] Glib_jll v2.82.4+0
  [3b182d85] Graphite2_jll v1.3.14+1
  [2e76f6c2] HarfBuzz_jll v8.5.0+0
  [1d5cc7b8] IntelOpenMP_jll v2024.2.1+0
  [aacddb02] JpegTurbo_jll v3.1.0+1
  [c1c5ebd0] LAME_jll v3.100.2+0
  [88015f11] LERC_jll v4.0.0+2
  [1d63c593] LLVMOpenMP_jll v18.1.7+0
  [dd4b983a] LZO_jll v2.10.2+2
⌅ [e9f186c6] Libffi_jll v3.2.2+2
  [d4300ac3] Libgcrypt_jll v1.11.0+0
  [7e76a0d4] Libglvnd_jll v1.7.0+0
  [7add5ba3] Libgpg_error_jll v1.51.0+0
  [94ce4f54] Libiconv_jll v1.17.0+1
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  [38a345b3] Libuuid_jll v2.40.2+1
  [856f044c] MKL_jll v2024.2.0+0
  [e7412a2a] Ogg_jll v1.3.5+1
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⌅ [30392449] Pixman_jll v0.43.4+0
  [c0090381] Qt6Base_jll v6.7.1+1
  [629bc702] Qt6Declarative_jll v6.7.1+2
  [ce943373] Qt6ShaderTools_jll v6.7.1+1
  [e99dba38] Qt6Wayland_jll v6.7.1+1
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  [2381bf8a] Wayland_protocols_jll v1.36.0+0
  [02c8fc9c] XML2_jll v2.13.5+0
  [aed1982a] XSLT_jll v1.1.42+0
  [ffd25f8a] XZ_jll v5.6.3+1
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  [4f6342f7] Xorg_libX11_jll v1.8.6+1
  [0c0b7dd1] Xorg_libXau_jll v1.0.11+1
  [935fb764] Xorg_libXcursor_jll v1.2.3+0
  [a3789734] Xorg_libXdmcp_jll v1.1.4+1
  [1082639a] Xorg_libXext_jll v1.3.6+1
  [d091e8ba] Xorg_libXfixes_jll v6.0.0+0
  [a51aa0fd] Xorg_libXi_jll v1.8.2+0
  [d1454406] Xorg_libXinerama_jll v1.1.5+0
  [ec84b674] Xorg_libXrandr_jll v1.5.4+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.11+1
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.1+1
  [c7cfdc94] Xorg_libxcb_jll v1.17.0+1
  [cc61e674] Xorg_libxkbfile_jll v1.1.2+1
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+0
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [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 v1.4.6+1
  [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
  [c5fb5394] Xorg_xtrans_jll v1.5.0+1
  [8f1865be] ZeroMQ_jll v4.3.5+2
  [3161d3a3] Zstd_jll v1.5.6+2
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  [1317d2d5] oneTBB_jll v2021.12.0+0
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
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  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [3fa0cd96] REPL
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  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
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  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
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  [bea87d4a] SuiteSparse_jll v7.2.1+1
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  [8e850b90] libblastrampoline_jll v5.8.0+1
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  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`