Evolution of the relative error in the conserved energy (top panels) and angular momentum (bottom panels) for the regular orbit RSA5 from Section 4.1.1 (left panels) and the E = 0.89, L = −7 chaotic orbit from Section 4.1.2 (right panels), using interpolated values of the electromagnetic fields. The simulation was run on a 64 × 64 × 128 grid with Δt = 1 until t = 100,000. The modified Hamiltonian scheme (magenta lines) preserves the energy to machine precision (∼10⁻¹⁵ in our simulation) in case of vanishing electric fields (top left panel), while the original Hamiltonian scheme (green lines) loses its energy-conserving character due to the interpolation step. In all cases, the RK4 scheme introduces a secular unbounded growth in the energy error, while the IMR scheme keeps energy errors bounded and with the same accuracy as the Hamiltonian schemes. The original Hamiltonian method retains exact momentum conservation in all cases, while the IMR and modified Hamiltonian schemes perform equally well in keeping errors in L bounded in time.