A Higher Order Finite Difference Method for Numerical Solution of the Kuramoto–Sivashinsky Equation

Abstract

The Kuramoto–Sivashinsky equation is a fundamental fourth-order partial differential equation modeling various nonlinear physical phenomena in unstable systems. It occupies a considerable position in explaining the motion of a fluid going down a vertical wall, a spatially uniform oscillating chemical reaction in a homogeneous medium and unstable drift waves in plasmas. The analytical treatment of this nonlinear differential equation is too involved a process and requires application of advanced mathematical tools, so it is required to develop efficient numerical techniques whose solutions are of great significance to scientists and engineers. One way of solving this equation is the application of compact finite difference method which is steadily acquiring popularity owing to its high accuracy and easy implementation. In this paper, a novel two-level implicit compact finite difference method to the solution of the one-dimensional Kuramoto–Sivashinsky equation subject to appropriate initial and boundary conditions is presented using coupled approach. The method is fourth-order accurate in space and second-order accurate in time. It is based on only three-spatial grid points of a compact stencil without the need to discretize the boundary conditions. Computational results are presented to illustrate the applicability and efficiency of the proposed method.