USE OF TAYLOR’S SERIES EXPANSION IN DEVELOPING FINITE DIFFERENCE METHODS FOR SOLVING THE WAVE EQUATION

Abstract

Hyperbolic partial differential equations (PDEs) are used to model physical phenomena such as wave propagation. Among the simplest hyperbolic equations are the 1-D advection equation which is a first order PDE and the 1-D wave equation which is a second order PDE. In this study 1-D hyperbolic partial second order differential equation was considered. Taylor’s series was used to expand the left hand side (the partial derivative of the solution u with respect to time t) of it at a point. The resulting partial derivatives were approximated using finite differences. Among the finite difference methods (FDMs) used were the explicit central time central space (CTCS) method and implicit methods. We applied these methods to 1-D wave equation. We discovered that the explicit CTCS scheme produced near exact results when the first set of terms in the Taylor’s series expansion were employed, but beyond this first set of terms the solution drifted further away from the exact. The implicit methods improved in accuracy as the numbers of terms in the Taylor’s series were employed.