We derive and analyze numerical methods for weak approximation of underdamped (kinetic) Langevin dynamics in bounded domains. First-order methods are based on an Euler-type scheme interlaced with collisions with the boundary. To achieve second order, composition schemes are derived based on decomposition of the generator into collisional drift, impulse, and stochastic momentum evolution. In a deterministic setting, this approach would typically lead to first-order approximation, even in symmetric compositions, but we find that the stochastic method can provide second-order weak approximation with a single gradient evaluation, both at finite times and in the ergodic limit. We provide theoretical and numerical justification for this observation using model problems and compare and contrast the numerical performance of different choices of the ordering of the terms in the splitting scheme.

翻译：我们推导并分析了有界域中欠阻尼（动力学）朗之万动力学弱近似的数值方法。一阶方法基于与边界碰撞交织的欧拉型方案。为实现二阶精度，我们基于将生成元分解为碰撞漂移、冲量和随机动量演化三项，构造了组合方案。在确定性背景下，即便采用对称组合，该方法通常也只能达到一阶近似精度，但研究发现该随机方法通过单次梯度求值即可实现二阶弱近似——无论是在有限时间尺度还是遍历极限情形下。我们通过模型问题对这一发现提供了理论和数值验证，并对分裂方案中不同项序的数值表现进行了比较与对比分析。