An improved curvilinear finite difference (CFD) method for arbitrary mesh systems

Abstract

This paper describes the automatic computer generation of finite difference approximations for partial derivatives in arbitrary mesh systems using an “Improved CFD” method. In the first part of the paper, the algorithm of the proposed Improved CFD method, used to solve a general second-order linear partial differential equation defined in a two-dimensional domain, will be presented. The performance of this numerical method will then be compared with the “Original CFD” method developed by Lau[1], and a so-called least square surface fit method developed by Liszka and Orkisz[2]. In the proposed method, extensive use is made of matrix algebra. The method is therefore highly systematic and can be easily implemented into computer programmes. Numerical examples tested in this paper indicate that, for irregular meshes, better numerical accuracy can be attained by the proposed method as compared with the Original CFD method. At the same time, the straightforward extension of the present method to the generation of higher-order-than-two finite difference approximations and to the solution of three-dimensional field problems will also be demonstrated. In the second part of the paper, the computational procedures for the numerical solution of a first boundaryvalue problem which is governed by a single second-order nonlinear partial differential equation will be derived.