A CA Computational Finite Difference Treatment for PDEs Including the Mixed Derivative Term with High Accuracy on Curved Domains"omputational Finite Difference Treatment for PDEs

Abstract

In this paper, a ¯nite di®erence treatment for Partial Di®erential Equa- tions (PDEs) with the mixed derivative term is described. The method depends on using a simple ¯rst order PDE for a new dependent variable. The method deals with any problem formulated by a single elliptic PDE or by an elliptic system of two PDEs. Applying this approach to problems with curved boundaries and regular regions will decrease the number of unknowns in each equation and at the same time will increase the number of algebraic equations linearly and the accuracy quadratically. Moreover, the consistency of the ¯nite di®erence representation of the system is achieved. Also, the derived system is of the same type as the original one. Two numerical applications are given. An e±cient numerical algorithm is designed.