An efficient algorithm for the numerical solution of telegraph interface model with discontinuous coefficients via Haar wavelets

Abstract

In this article, a hybrid numerical method based on Haar wavelets and finite difference method is proposed for the solution of hyperbolic telegraph interface model in one spacial dimension. We considered the problems having both constant and variable coefficients around interfaces of discontinuity across a fixed interface. In this method, the highest order spatial derivative is approximated by truncated Haar series, while for temporal derivative finite difference method is utilized. The proposed method is applied to some benchmark linear and nonlinear test problems. The exact and approximate solutions are compared for different number of collocation points at different time steps. The algorithm based on this new method is simple and can be easily implemented in any programming language. Experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results. The proposed method perform very well and produces a stable solution if sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, whereas, the other existing method loses its accuracy in such cases. The numerical experiments, stability and rate of convergence both theoretical and computational, confirm the accuracy and diverse applicability of the method.

Keywords

• Hyperbolic telegraph equation
• Interface models
• Haar wavelet
• Collocation method
• Finite difference method