This paper investigates the use of the localized method of fundamental solutions (LMFS) for the numerical solution of general transient convection-diffusion-reaction equation in both two-(2D) and three-dimensional (3D) materials. The method is developed as a generalization of the author's earlier work on Laplace's equation to transient convection-diffusion-reaction equation. The popular Crank-Nicolson (CN) time-stepping technology is adopted to perform the temporal simulations. The LMFS approach is then introduced for solving the resulting inhomogeneous boundary value problems, where a pseudo-spectral Chebyshev collocation scheme (CCS) is employed for the approximation of the corresponding particular solutions. As compared with the classical MFS and boundary element method (BEM), the present CN-CCS-LMFS approach produces sparse and banded stiffness matrix which makes the method possible to perform large-scale dynamic simulations. Several benchmark numerical examples are presented to demonstrate the efficiency and feasibility of the present method.

本文研究了基本解的局部化方法（LMFS）在二维（2D）和三维（3D）材料中通用瞬态对流扩散反应方程的数值解中的应用。该方法是作者先前将Laplace方程的早期工作推广到瞬态对流-扩散-反应方程的概括。采用流行的Crank-Nicolson（CN）时步技术进行时间仿真。然后引入LMFS方法来解决由此产生的不均匀边值问题，其中采用伪谱Chebyshev配置方案（CCS）来近似对应的特定解决方案。与经典的MFS和边界元方法（BEM）相比，当前的CN-CCS-LMFS方法生成稀疏带状刚度矩阵，这使得该方法可以执行大规模动态仿真。给出了几个基准数值示例，以证明本方法的效率和可行性。