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.

中文翻译：

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具有间断势的哈密顿力学系统的精确有效仿真

本文考虑具有潜在功能的哈密顿机械系统，其允许跳跃不连续性。重点是解决方案的精确有效数值近似，这将通过反射和折射定律定义。尽管辛积分器在光滑的机械系统中是成功的，但对于不连续的积分器而言，它们的构造是不平凡的，并且数值收敛阶也可能受到损害。提出了几种相当有用的数值方法，包括：用于一般问题的一阶辛积分器，仅具有一个线性界面的问题的三阶辛积分器，用于一般问题（不再辛）的任意高阶可逆积分，以及以前的高阶方法的自适应时步版本。有趣的是 由于不连续性，不清楚辛度是否会导致长时间的良好性能，因为传统的哈密顿量向后误差分析不再适用。因此，在此阶段，我们建议的默认方法是最后一个。提供了各种数值证据，包括收敛顺序，长时间性能，动量图守恒以及与计算昂贵的罚分方法的一致性。还提出了一个复杂的问题，即炒蘑菇，并对其进行了数值研究，观察到了被困和遍历动力学之间的多重分歧。提供了各种数值证据，包括收敛顺序，长时间性能，动量图守恒以及与计算昂贵的罚分方法的一致性。还提出了一个复杂的问题，即炒蘑菇，并对其进行了数值研究，观察到了被困和遍历动力学之间的多重分歧。提供了各种数值证据，包括收敛顺序，长时间性能，动量图守恒以及与计算昂贵的罚分方法的一致性。还提出了一个复杂的问题，即炒蘑菇，并对其进行了数值研究，观察到了被困和遍历动力学之间的多重分歧。