Sixth-order compact finite difference scheme with discrete sine transform for solving Poisson equations with Dirichlet boundary conditions

Abstract

Compact finite difference methods are very popular for solving differential equations that arise in a wide variety of real-world applications. Despite their popularity, the efficiency of these methods is limited by the need for matrix inversion which is troubling when the size of the matrix is very large. However, there is a growing demand for methods with better accuracy and computational speed. To meet such needs, we design a new method which uses a sixth-order compact finite difference scheme and a discrete sine transform to solve Poisson equations with Dirichlet boundary conditions. The scheme is developed using the concept of higher-order Taylor series expansion which is then used to discretize Poisson equations that results in a system of linear algebraic equations. To solve this system, we propose a discrete sine transform designed based on the developed compact finite difference scheme. We proved analytically and numerically that the order of convergence of the proposed method is six. The efficiency of the new scheme is demonstrated by solving different test problems. The numerical results indicate that the proposed method outperforms an existing fourth-order scheme and provides solutions that are in excellent agreement with the exact solution.