In this paper, we focus on numerical approximations of Piecewise Diffusion Markov Processes (PDifMPs), particularly when the explicit flow maps are unavailable. Our approach is based on the thinning method for modelling the jump mechanism and combines the Euler-Maruyama scheme to approximate the underlying flow dynamics. For the proposed approximation schemes, we study both the mean-square and weak convergence. Weak convergence of the algorithms is established by a martingale problem formulation. Moreover, we employ these results to simulate the migration patterns exhibited by moving glioma cells at the microscopic level. Further, we develop and implement a splitting method for this PDifMP model and employ both the Thinned Euler-Maruyama and the splitting scheme in our simulation example, allowing us to compare both methods.

翻译：本文聚焦于分段扩散马尔可夫过程的数值逼近方法，尤其针对显式流映射不可得的情形。我们的方法基于模拟跳跃机制的细化方法，并结合Euler-Maruyama格式来逼近底层流动力学。针对所提出的逼近格式，我们同时研究了均方收敛与弱收敛性。算法的弱收敛性通过鞅问题框架予以建立。此外，我们运用这些结果模拟了微观尺度下运动胶质瘤细胞所呈现的迁移模式。进一步地，我们针对该分段扩散马尔可夫过程模型开发并实现了分裂算法，并在仿真示例中同时采用细化Euler-Maruyama方法与分裂格式，从而实现对两种方法的比较分析。