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（PDifMPs）的数值逼近，特别是在显式流映射不可用的情况下。我们基于稀疏化方法对跳跃机制进行建模，并结合欧拉-丸山格式逼近底层流动力学。针对所提出的逼近格式，我们研究了均方收敛性与弱收敛性。通过鞅问题形式化建立了算法的弱收敛性。此外，我们运用这些结果在微观层面模拟移动胶质瘤细胞的迁移模式。进一步，我们针对该PDifMP模型开发并实现了一种分裂格式，并在仿真示例中同时采用稀疏化欧拉-丸山格式与分裂格式，以比较两种方法的效果。