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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)
48.981314 seconds (133.70 M allocations: 7.751 GiB, 9.09% gc time, 99.99% 
compilation time)
  1.888414 seconds (3.69 M allocations: 236.124 MiB, 99.57% compilation tim
e)
  2.052458 seconds (2.79 M allocations: 180.968 MiB, 2.24% gc time, 98.45% 
compilation time)
 28.570658 seconds (92.08 M allocations: 5.517 GiB, 10.02% gc time, 99.94% 
compilation time)
  2.752194 seconds (2.77 M allocations: 182.998 MiB, 99.87% compilation tim
e)
  2.076174 seconds (3.41 M allocations: 352.864 MiB, 4.06% gc time, 90.61% 
compilation time)
Vern9 + ManifoldProjection max energy error:	3.410605131648481e-13	in	317	s
teps.
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)
0.004913 seconds (40.70 k allocations: 66.917 MiB)
  0.073772 seconds (700.10 k allocations: 61.899 MiB)
  0.123392 seconds (700.10 k allocations: 61.901 MiB)
  0.176156 seconds (1.60 M allocations: 105.255 MiB)
  0.032021 seconds (372.11 k allocations: 17.920 MiB)
  1.586769 seconds (6.38 M allocations: 1.591 GiB, 9.85% gc time)
Vern9 + ManifoldProjection max energy error:	3.410605131648481e-13	in	317	s
teps.
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)
0.004582 seconds (40.70 k allocations: 66.917 MiB)
  1.821144 seconds (7.00 M allocations: 579.841 MiB, 52.77% gc time)
  1.962801 seconds (7.00 M allocations: 579.844 MiB, 33.08% gc time)
  2.001987 seconds (16.04 M allocations: 1.013 GiB, 16.16% gc time)
  0.331641 seconds (3.72 M allocations: 174.938 MiB)
 13.077320 seconds (63.82 M allocations: 15.904 GiB, 13.44% gc time)
Vern9 + ManifoldProjection max energy error:	3.410605131648481e-13	in	317	s
teps.
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)
0.006350 seconds (40.70 k allocations: 66.917 MiB, 21.74% gc time)
  4.056876 seconds (14.00 M allocations: 1.171 GiB, 56.30% gc time)
  3.111313 seconds (14.00 M allocations: 1.171 GiB, 15.54% gc time)
 15.157659 seconds (32.09 M allocations: 2.023 GiB, 77.47% gc time)
  0.636878 seconds (7.43 M allocations: 347.531 MiB)
 27.505710 seconds (127.64 M allocations: 31.807 GiB, 17.31% gc time)
Vern9 + ManifoldProjection max energy error:	3.410605131648481e-13	in	317	s
teps.
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:
  0.004868 seconds (40.70 k allocations: 66.917 MiB)
  0.073710 seconds (700.10 k allocations: 61.899 MiB)
  0.130010 seconds (700.10 k allocations: 61.901 MiB)
  0.175532 seconds (1.60 M allocations: 105.255 MiB)
  0.032317 seconds (372.11 k allocations: 17.920 MiB)
  1.579815 seconds (6.38 M allocations: 1.591 GiB, 8.64% gc time)
Vern9 + ManifoldProjection max energy error:	3.410605131648481e-13	in	317	s
teps.
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:
  1.345975 seconds (2.76 M allocations: 202.492 MiB, 2.30% gc time, 97.98% 
compilation time)
  0.702764 seconds (1.18 M allocations: 92.843 MiB, 93.72% compilation time
)
  1.441413 seconds (2.94 M allocations: 180.208 MiB, 2.44% gc time, 98.26% 
compilation time)
  0.841186 seconds (1.14 M allocations: 78.429 MiB, 98.64% 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:
  0.007316 seconds (70.09 k allocations: 5.806 MiB)
  0.011831 seconds (70.09 k allocations: 5.808 MiB)
  0.014336 seconds (160.71 k allocations: 10.781 MiB)
  0.003074 seconds (37.49 k allocations: 1.872 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:
  0.002943 seconds (10.02 k allocations: 1.681 MiB)
  0.004452 seconds (10.02 k allocations: 1.682 MiB)
  0.002132 seconds (4.90 k allocations: 2.762 MiB)
  0.001299 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:
  0.073080 seconds (700.10 k allocations: 61.899 MiB)
  0.127132 seconds (700.10 k allocations: 61.901 MiB)
  0.175429 seconds (1.60 M allocations: 105.255 MiB)
  0.032445 seconds (372.11 k allocations: 17.920 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:
  0.027652 seconds (100.03 k allocations: 21.886 MiB)
  0.044136 seconds (100.03 k allocations: 21.887 MiB)
  0.019910 seconds (48.67 k allocations: 23.950 MiB)
  0.010986 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:
  1.840837 seconds (7.00 M allocations: 579.841 MiB, 53.43% gc time)
  1.313025 seconds (7.00 M allocations: 579.844 MiB)
  2.038694 seconds (16.04 M allocations: 1.013 GiB, 16.75% gc time)
  0.320925 seconds (3.72 M allocations: 174.938 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:
  0.312440 seconds (1.00 M allocations: 167.849 MiB, 5.51% gc time)
  0.481468 seconds (1.00 M allocations: 167.850 MiB)
  0.244364 seconds (486.25 k allocations: 215.754 MiB)
  0.111061 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:
  2.874783 seconds (14.00 M allocations: 1.171 GiB, 39.47% gc time)
  4.579002 seconds (14.00 M allocations: 1.171 GiB, 42.98% gc time)
  9.367741 seconds (32.09 M allocations: 2.023 GiB, 63.79% gc time)
  0.636116 seconds (7.43 M allocations: 347.531 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:
  0.756659 seconds (2.00 M allocations: 395.141 MiB, 13.61% gc time)
  1.175719 seconds (2.00 M allocations: 395.141 MiB, 14.31% gc time)
  0.489461 seconds (972.45 k allocations: 428.591 MiB)
  0.220328 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 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 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-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DynamicalODE/Project.toml`
  [459566f4] DiffEqCallbacks v4.6.0
  [055956cb] DiffEqPhysics v3.15.0
  [b305315f] Elliptic v1.0.1
  [1dea7af3] OrdinaryDiffEq v6.98.0
  [65888b18] ParameterizedFunctions v5.18.0
  [91a5bcdd] Plots v1.40.13
  [31c91b34] SciMLBenchmarks v0.1.3
  [90137ffa] StaticArrays v1.9.13
  [92b13dbe] TaylorIntegration v0.17.1
  [37e2e46d] LinearAlgebra
  [de0858da] Printf
  [10745b16] Statistics v1.10.0

And the full manifest:

Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DynamicalODE/Manifest.toml`
  [47edcb42] ADTypes v1.14.0
  [1520ce14] AbstractTrees v0.4.5
  [7d9f7c33] Accessors v0.1.42
  [79e6a3ab] Adapt v4.3.0
  [66dad0bd] AliasTables v1.1.3
  [ec485272] ArnoldiMethod v0.4.0
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  [4c555306] ArrayLayouts v1.11.1
⌅ [e2ed5e7c] Bijections v0.1.10
  [d1d4a3ce] BitFlags v0.1.9
  [62783981] BitTwiddlingConvenienceFunctions v0.1.6
  [8e7c35d0] BlockArrays v1.6.3
  [70df07ce] BracketingNonlinearSolve v1.3.0
  [2a0fbf3d] CPUSummary v0.2.6
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  [fb6a15b2] CloseOpenIntervals v0.1.13
  [944b1d66] CodecZlib v0.7.8
  [35d6a980] ColorSchemes v3.29.0
  [3da002f7] ColorTypes v0.12.1
  [c3611d14] ColorVectorSpace v0.11.0
  [5ae59095] Colors v0.13.1
  [861a8166] Combinatorics v1.0.3
  [a80b9123] CommonMark v0.9.1
  [38540f10] CommonSolve v0.2.4
  [bbf7d656] CommonSubexpressions v0.3.1
  [f70d9fcc] CommonWorldInvalidations v1.0.0
  [34da2185] Compat v4.16.0
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  [a33af91c] CompositionsBase v0.1.2
  [2569d6c7] ConcreteStructs v0.2.3
  [f0e56b4a] ConcurrentUtilities v2.5.0
  [8f4d0f93] Conda v1.10.2
  [187b0558] ConstructionBase v1.5.8
  [d38c429a] Contour v0.6.3
  [adafc99b] CpuId v0.3.1
  [a8cc5b0e] Crayons v4.1.1
  [9a962f9c] DataAPI v1.16.0
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  [e2d170a0] DataValueInterfaces v1.0.0
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  [2b5f629d] DiffEqBase v6.175.0
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  [77a26b50] DiffEqNoiseProcess v5.24.1
  [055956cb] DiffEqPhysics v3.15.0
  [163ba53b] DiffResults v1.1.0
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⌅ [a0c0ee7d] DifferentiationInterface v0.6.54
  [8d63f2c5] DispatchDoctor v0.4.19
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  [b305315f] Elliptic v1.0.1
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⌃ [f151be2c] EnzymeCore v0.8.9
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  [a4df4552] FastPower v1.1.2
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  [64ca27bc] FindFirstFunctions v1.4.1
  [6a86dc24] FiniteDiff v2.27.0
  [53c48c17] FixedPointNumbers v0.8.5
  [1fa38f19] Format v1.3.7
⌅ [f6369f11] ForwardDiff v0.10.38
  [069b7b12] FunctionWrappers v1.1.3
  [77dc65aa] FunctionWrappersWrappers v0.1.3
  [d9f16b24] Functors v0.5.2
  [46192b85] GPUArraysCore v0.2.0
  [28b8d3ca] GR v0.73.16
  [c145ed77] GenericSchur v0.5.5
  [d7ba0133] Git v1.4.0
  [c27321d9] Glob v1.3.1
  [86223c79] Graphs v1.12.1
  [42e2da0e] Grisu v1.0.2
  [cd3eb016] HTTP v1.10.16
  [eafb193a] Highlights v0.5.3
  [34004b35] HypergeometricFunctions v0.3.28
⌃ [7073ff75] IJulia v1.27.0
  [615f187c] IfElse v0.1.1
  [3263718b] ImplicitDiscreteSolve v0.1.2
  [d25df0c9] Inflate v0.1.5
  [18e54dd8] IntegerMathUtils v0.1.2
  [8197267c] IntervalSets v0.7.11
  [3587e190] InverseFunctions v0.1.17
  [92d709cd] IrrationalConstants v0.2.4
  [82899510] IteratorInterfaceExtensions v1.0.0
  [1019f520] JLFzf v0.1.11
  [692b3bcd] JLLWrappers v1.7.0
  [682c06a0] JSON v0.21.4
  [ae98c720] Jieko v0.2.1
  [98e50ef6] JuliaFormatter v2.1.2
⌅ [70703baa] JuliaSyntax v0.4.10
  [ccbc3e58] JumpProcesses v9.15.0
  [ba0b0d4f] Krylov v0.10.1
  [b964fa9f] LaTeXStrings v1.4.0
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  [5078a376] LazyArrays v2.6.1
  [87fe0de2] LineSearch v0.1.4
  [d3d80556] LineSearches v7.3.0
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  [739be429] MbedTLS v1.1.9
  [442fdcdd] Measures v0.3.2
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  [961ee093] ModelingToolkit v10.0.1
  [2e0e35c7] Moshi v0.3.5
  [46d2c3a1] MuladdMacro v0.2.4
  [102ac46a] MultivariatePolynomials v0.5.9
  [ffc61752] Mustache v1.0.20
  [d8a4904e] MutableArithmetics v1.6.4
  [d41bc354] NLSolversBase v7.9.1
  [77ba4419] NaNMath v1.1.3
  [8913a72c] NonlinearSolve v4.9.0
  [be0214bd] NonlinearSolveBase v1.12.0
  [5959db7a] NonlinearSolveFirstOrder v1.5.0
  [9a2c21bd] NonlinearSolveQuasiNewton v1.6.0
  [26075421] NonlinearSolveSpectralMethods v1.2.0
  [6fe1bfb0] OffsetArrays v1.17.0
  [4d8831e6] OpenSSL v1.5.0
  [429524aa] Optim v1.12.0
  [bac558e1] OrderedCollections v1.8.1
  [1dea7af3] OrdinaryDiffEq v6.98.0
  [89bda076] OrdinaryDiffEqAdamsBashforthMoulton v1.2.0
  [6ad6398a] OrdinaryDiffEqBDF v1.5.1
  [bbf590c4] OrdinaryDiffEqCore v1.26.0
  [50262376] OrdinaryDiffEqDefault v1.4.0
  [4302a76b] OrdinaryDiffEqDifferentiation v1.9.1
  [9286f039] OrdinaryDiffEqExplicitRK v1.1.0
  [e0540318] OrdinaryDiffEqExponentialRK v1.4.0
  [becaefa8] OrdinaryDiffEqExtrapolation v1.5.0
  [5960d6e9] OrdinaryDiffEqFIRK v1.12.0
  [101fe9f7] OrdinaryDiffEqFeagin v1.1.0
  [d3585ca7] OrdinaryDiffEqFunctionMap v1.1.1
  [d28bc4f8] OrdinaryDiffEqHighOrderRK v1.1.0
  [9f002381] OrdinaryDiffEqIMEXMultistep v1.3.0
  [521117fe] OrdinaryDiffEqLinear v1.3.0
  [1344f307] OrdinaryDiffEqLowOrderRK v1.2.0
  [b0944070] OrdinaryDiffEqLowStorageRK v1.3.0
  [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.10.0
  [c9986a66] OrdinaryDiffEqNordsieck v1.1.0
  [5dd0a6cf] OrdinaryDiffEqPDIRK v1.3.1
  [5b33eab2] OrdinaryDiffEqPRK v1.1.0
  [04162be5] OrdinaryDiffEqQPRK v1.1.0
  [af6ede74] OrdinaryDiffEqRKN v1.1.0
  [43230ef6] OrdinaryDiffEqRosenbrock v1.10.1
  [2d112036] OrdinaryDiffEqSDIRK v1.3.0
  [669c94d9] OrdinaryDiffEqSSPRK v1.3.0
  [e3e12d00] OrdinaryDiffEqStabilizedIRK v1.3.0
  [358294b1] OrdinaryDiffEqStabilizedRK v1.1.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v1.3.0
  [b1df2697] OrdinaryDiffEqTsit5 v1.1.0
  [79d7bb75] OrdinaryDiffEqVerner v1.2.0
  [90014a1f] PDMats v0.11.35
  [65888b18] ParameterizedFunctions v5.18.0
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  [69de0a69] Parsers v2.8.3
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  [1d0040c9] PolyesterWeave v0.2.2
  [85a6dd25] PositiveFactorizations v0.2.4
  [d236fae5] PreallocationTools v0.4.27
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.4.3
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  [7e49a35a] RuntimeGeneratedFunctions v0.5.15
  [9dfe8606] SCCNonlinearSolve v1.2.0
  [94e857df] SIMDTypes v0.1.0
  [0bca4576] SciMLBase v2.96.0
  [31c91b34] SciMLBenchmarks v0.1.3
  [19f34311] SciMLJacobianOperators v0.1.6
⌃ [c0aeaf25] SciMLOperators v0.4.0
  [431bcebd] SciMLPublic v1.0.0
  [53ae85a6] SciMLStructures v1.7.0
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  [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.2.0
  [727e6d20] SimpleNonlinearSolve v2.5.0
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  [b85f4697] SoftGlobalScope v1.1.0
  [a2af1166] SortingAlgorithms v1.2.1
  [0a514795] SparseMatrixColorings v0.4.20
  [276daf66] SpecialFunctions v2.5.1
  [860ef19b] StableRNGs v1.0.3
  [aedffcd0] Static v1.2.0
  [0d7ed370] StaticArrayInterface v1.8.0
  [90137ffa] StaticArrays v1.9.13
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  [82ae8749] StatsAPI v1.7.1
  [2913bbd2] StatsBase v0.34.5
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  [6e34b625] Bzip2_jll v1.0.9+0
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⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
  [a3f928ae] Fontconfig_jll v2.16.0+0
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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`