Evolution of the relative error in the conserved energy (top panels) and angular momentum (bottom panels) for the E = 0.885, L = −7 regular orbit (left panels) and the E = 0.89, L = −7 chaotic orbit (right panels). The IMR (blue lines) and modified Hamiltonian (magenta lines) schemes keep errors in energy and momentum bounded at all times. The original Hamiltonian scheme (green lines) preserves both quantities to machine precision, while the RK4 scheme (red lines) introduces a secular unbounded growth in the error leading to the spurious release of the particle from the orbit.