Influence of the Forward Difference Scheme for the Time Derivative on the Stability of Wave Equation Numerical Solution

TL;DR

This paper investigates how the forward difference scheme for the time derivative affects the stability of numerical solutions to the wave equation, comparing it with other schemes and highlighting the importance of spatial discretization.

Contribution

It provides a detailed stability analysis of the forward difference scheme in the wave equation context, showing stability depends on both time and spatial discretization schemes.

Findings

Forward difference scheme does not always cause instability.

Stability can be maintained with appropriate spatial difference schemes.

The choice of spatial scheme is crucial for overall stability.

Abstract

Research on numerical stability of difference equations has been quite intensive in the past century. The choice of difference schemes for the derivative terms in these equations contributes to a wide range of the stability analysis issues - one of which is how a chosen scheme may directly or indirectly contribute to such stability. In the present paper, how far the forward difference scheme for the time derivative in the wave equation influences the stability of the equation numerical solution, is particularly investigated. The stability analysis of the corresponding difference equation involving four schemes, namely Lax's, central, forward, and rearward differences, were carried out, and the resulting stability criteria were compared. The results indicate that the instability of the solution of wave equation is not always due to the forward difference scheme for the time derivative. Rather, it is shown in this paper that the stability criterion is still possible when the spatial derivative is represented by an appropriate difference scheme. This sheds light on the degree of applicability of a difference scheme for a hyperbolic equation.