Accurate and Efficient Simulations of Hamiltonian Mechanical Systems with Discontinuous Potentials

TL;DR

This paper develops and compares numerical methods for simulating Hamiltonian systems with discontinuous potentials, addressing challenges in accuracy and efficiency, and providing evidence of their performance through various numerical experiments.

Contribution

It introduces new symplectic and high-order integrators tailored for discontinuous potentials in Hamiltonian systems, expanding the toolkit beyond traditional smooth-system methods.

Findings

High-order reversible integrators perform well for discontinuous potentials.

Symplecticity's advantage diminishes due to discontinuities.

The adaptive high-order method is recommended as default.

Abstract

This article considers Hamiltonian mechanical systems with potential functions admitting jump discontinuities. The focus is on accurate and efficient numerical approximations of their solutions, which will be defined via the laws of reflection and refraction. Despite of the success of symplectic integrators for smooth mechanical systems, their construction for the discontinuous ones is nontrivial, and numerical convergence order can be impaired too. Several rather-usable numerical methods are proposed, including: a first-order symplectic integrator for general problems, a third-order symplectic integrator for problems with only one linear interface, arbitrarily high-order reversible integrators for general problems (no longer symplectic), and an adaptive time-stepping version of the previous high-order method. Interestingly, whether symplecticity leads to favorable long time performance is no longer clear due to discontinuity, as traditional Hamiltonian backward error analysis does not apply any more. Therefore, at this stage, our recommended default method is the last one. Various numerical evidence, on the order of convergence, long time performance, momentum map conservation, and consistency with the computationally-expensive penalty method, are supplied. A complex problem, namely the Sauteed Mushroom, is also proposed and numerically investigated, for which multiple bifurcations between trapped and ergodic dynamics are observed.