# Questions on the DynamicalODE integrators

Hi!

I have a question about the integrators in the differential equation community. The integrators for `DynamicalODEProblem` are listed in Dynamical, Hamiltonian, and 2nd Order ODE Solvers · DifferentialEquations.jl. While testing for the integrators, especially those symplectic ones, I found some puzzling results regarding energy conservation:

``````# Tracing charged particle in a static EM field.

using TestParticle
using OrdinaryDiffEq
using StaticArrays
using LinearAlgebra: ×
#using Plots

const B₀ = 1e-8 # [T]

function location!(dx, x, v, p::TestParticle.TPTuple, t)
dx .= v
end

function lorentz!(dv, x, v, p::TestParticle.TPTuple, t)
q, m, E, B = p
dv .= q/m*(E(x, t) + v × (B(x, t)))
end

## Initialize field

function uniform_B(x)
return SA[0, 0, B₀]
end

function uniform_E(x)
return SA[0.0, 0.0, 0.0]
end

"Check energy conservation."
function getenergy(sol)
vx, vy, vz = sol.u[1][4:6]
KE1 = √(vx^2 + vy^2 + vz^2)
vx, vy, vz = sol.u[end][4:6]
KE2 = √(vx^2 + vy^2 + vz^2)
KE1, KE2
end

## Initialize particles

x0 = [0.0, 0, 0]
v0 = [0.0, 1e2, 0.0]
stateinit = [x0..., v0...]
tspan_proton = (0.0, 2000.0)

param_proton = prepare(uniform_E, uniform_B, species=Proton)

## Solve for the trajectories

prob_p = DynamicalODEProblem(location!, lorentz!, x0, v0, tspan_proton, param_proton)

Ωᵢ = TestParticle.qᵢ * B₀ / TestParticle.mᵢ
Tᵢ = 2π / Ωᵢ
println("Number of gyrations: ", tspan_proton[2] / Tᵢ)

function main(prob_p)
@info "Tsit5"
@time sol_p = solve(prob_p, Tsit5(); save_idxs=[1,2,3,4,5,6]);

KE1, KE2 = getenergy(sol_p)
println("Energy: ", KE1, " to ", round(KE2, digits=4))
@info "KahanLi6"
@time sol_p = solve(prob_p, KahanLi6(); dt=0.01, save_idxs=[1,2,3,4,5,6]);

KE1, KE2 = getenergy(sol_p)
println("Energy: ", KE1, " to ", round(KE2, digits=4))
@info "Trapezoid"
@time sol_p = solve(prob_p, Trapezoid(); save_idxs=[1,2,3,4,5,6]);

KE1, KE2 = getenergy(sol_p)
println("Energy: ", KE1, " to ", round(KE2, digits=4))
@info "Vern6"
@time sol_p = solve(prob_p, Vern6(); save_idxs=[1,2,3,4,5,6]); # higher accuracy than Tsit5
KE1, KE2 = getenergy(sol_p)
println("Energy: ", KE1, " to ", round(KE2, digits=4))
end

main(prob_p)
#=
# Visualization
xyz = plot(sol_p, vars=(1,2,3), xlabel="x", ylabel="y", zlabel="z",
aspect_ratio=:equal, label="proton", lw=2)
=#
``````
``````julia> main(prob_p);
[ Info: Tsit5
0.010167 seconds (61.31 k allocations: 3.842 MiB)
Energy: 100.0 to 109.8673
[ Info: KahanLi6
0.458471 seconds (2.80 M allocations: 237.040 MiB, 50.05% gc time)
Energy: 100.0 to 887502.8753
[ Info: Trapezoid
0.082830 seconds (1.05 M allocations: 25.859 MiB)
Energy: 100.0 to 100.5663
[ Info: Vern6
0.004061 seconds (51.38 k allocations: 3.406 MiB)
Energy: 100.0 to 97.2829
``````

This shows that the symplectic integrator `KahanLi6` does not conserve energy in my case, while others perform reasonably well (< 10% energy change) for a single proton that gyrates for 300 periods in a static magnetic field. Did I set something wrong when using the symplectic integrators?

Symplectic integrators do not conserve energy. Though that looks like an instability: does decreasing dt help?

Yes decreasing dt helps (`dt=0.01` in the original post):

``````[ Info: KahanLi6
# dt = 0.001
Energy: 100.0 to 248.2101
# dt = 0.0001
Energy: 100.0 to 109.5171
``````

With `Tsit5` the time step quickly grows from 7.2233651437407934e-6 in the first step to about 0.5 in the 10th step.

Okay, then that might’ve just been above the stability limit on KahanLi6.

However it seems that the time step limit for all listed dynamical ODE integrators is pretty small compared with `Tsit5()` or other general methods? Is that expected?

Somewhat. It really depends on the kind of problem being solved but indeed some explicit methods have been designed with a larger or more robust stability region in mind.