A meshless Random Walk on arbitrary parallelepipeds simulation algorithm is developed and implemented for solving transient anisotropic diffusion-reaction equations. In contrast to the conventional Feynman-Kac based algorithm the suggested method does not use small time step simulations of the relevant diffusion processes. Instead, exact simulation of large random jumps over a set of appropriately constructed parallelepipeds in the domain is carried out. This decreases the cost of simulations considerably especially for domains with complicated boundary shape. Application to the problem of time-resolved cathodoluminescence intensity calculations for semiconductor materials with a set of threading dislocations is given. Important issues are the construction of an efficient sampling method from the first passage time density and the position distribution on the surface of an arbitrary parallelepiped. We combine a rejection algorithm and a probability density tabulation approach to construct optimal sampling methods from different densities including the random time a particle spends in a parallelepiped before it is absorbed inside it. We present in the last section results of computer simulation for the evaluation of the exciton flux to dislocations and a plane substrate, the cathodoluminescence intensity for threading dislocations imaging, and the concentration of the survived excitons. In addition, to validate the developed algorithms we have compared the computer simulations with the exact results, and obtained a perfect agreement.

开发并实现了一种任意平行六面体上的无网格随机游走模拟算法，用于求解瞬态各向异性扩散反应方程。与传统的基于 Feynman-Kac 的算法相比，建议的方法不使用相关扩散过程的小时间步模拟。相反，在域中一组适当构造的平行六面体上进行大随机跳跃的精确模拟。这大大降低了模拟成本，特别是对于具有复杂边界形状的域。给出了对具有一组螺纹位错的半导体材料的时间分辨阴极发光强度计算问题的应用。重要的问题是从第一次通过时间密度和任意平行六面体表面上的位置分布构建有效的采样方法。我们结合拒绝算法和概率密度制表方法来构建不同密度的最佳采样方法，包括粒子在被吸收之前在平行六面体中花费的随机时间。我们在最后一节介绍了计算机模拟的结果，用于评估位错和平面衬底的激子通量、穿线位错成像的阴极发光强度以及幸存激子的浓度。此外，为了验证所开发的算法，我们将计算机模拟与精确结果进行了比较，并获得了完美的一致性。