FDMFS for diffusion equation with unsteady forcing function

Abstract

In this paper, a novel numerical scheme called (FDMFS), which combines the finite difference method (FDM) and the method of fundamental solutions (MFS), is proposed to simulate the nonhomogeneous diffusion problem with an unsteady forcing function. Most mesh-less methods are confined to the investigations of nonhomogeneous diffusion equations with steady forcing functions due to the difficulty to find an unsteady particular solution. Therefore, we proposed a FDM with Cartesian grid to handle the unsteady nonhomogeneous term of the equations. The numerical solution in FDMFS is decomposed into a particular solution and a homogeneous solution. The particular solution is constructed using the FDM in an artificial regular domain which contains the real irregular domain without boundary conditions, and the homogeneous solution can be obtained by the time-space unification MFS in the irregular domain with boundary conditions. Besides, the Cartesian grid for particular solution is very simple to generate automatically. Our paper is the first time to propose an algorithm to solve nonhomogeneous diffusion equations with unsteady forcing functions using MFS to solve homogeneous solutions and FDM to calculate the particular solutions. Numerical experiments are presented for 2D problems in regular and irregular domains to show the high performance of this proposed scheme. Moreover, the stabilities of explicit and implicit FDM for particular solution are analyzed. Numerical studies suggest that the proposed FDMFS can speed up the simulation and save the CPU time and memory storage substantially.