Parallel Numerical Solution of Linear PDEs Using Implicit and Explicit Finite Difference Methods

Abstract

This paper presents the parallel computation of numerical solution of 3D linear PDEs discretized by using the Finite Difference Method (FDM). The parallel computation of numerical solution is carried out by two different solution methods. First, a typical discretized PDE is expressed as an implicit function of unknowns that lead to a system of linear algebraic equations represented in standard matrix form as AX=B. Next, the discretized PDE is explicitly defined for the grid points lying on the computational domain. Both the implicit and explicit forms of the problem are parallelized by using data parallelism technique. The main objective of this study is to analyze the computational time (sec) and memory (Mb) requirements for the implicit (matrix based) and explicit (matrix free) methods. For the testing and implementation purpose a typical 3D Poisson’s equation with Dirichlet boundary conditions is used. The parallel methods are executed on MATLAB parallel computing environment. The results revealed that the implicit solution method uses huge amount of computer memory than the explicit approach. Also, the parallel computational time is higher in former than later. This concludes that the explicit method offers good performance as compared to implicit method implemented on parallel system.