A fourth-order finite difference method based on spline in tension approximation for the solution of one-space dimensional second-order quasi-linear hyperbolic equations

Ranjan K Mohanty,Venu Gopal

doi:10.1186/1687-1847-2013-70

Abstract

In this paper, we propose a new three-level implicit nine-point compact finite difference formulation of order two in time and four in space directions, based on spline in tension approximation in x-direction and central finite difference approximation in t-direction for the numerical solution of one-space dimensional second-order quasi-linear hyperbolic equations with first-order space derivative term. We describe the mathematical formulation procedure in detail and also discuss how our formulation is able to handle a wave equation in polar coordinates. The proposed method, when applied to a general form of the telegrapher equation, is also shown to be unconditionally stable. Numerical examples are used to illustrate the usefulness of the proposed method. MSC: 65M06, 65M12.

Highlights

We consider the one-space dimensional second-order quasi-linear hyperbolic equation ∂ u∂t = A(x, t, u) ∂x + g(x, t, u, ux, ut), < x ( . )with the following initial conditions: u(x, ) = a(x), ut(x, ) = b(x), ≤ x ≤and the boundary conditions u(, t) = p (t), u(, t) = p (t), t ≥ .We assume that the conditions ( . ) and ( . ) are given with sufficient smoothness to maintain the order of accuracy in the numerical method under consideration.The study of a second-order quasilinear hyperbolic equation is of keen interest in the fields like acoustics, electromagnetics, fluid dynamics, mathematical physics, engineering etc
In Section, we propose a new three-level Numerov-type finite difference method based on spline in tension approximation
3 The finite difference method based on spline in tension approximation For the sake of the simplicity, first we consider the one-space dimensional nonlinear hyperbolic partial differential equation

Summary

Introduction

We follow the idea of Jain et al [ , ] but use non-polynomial tension spline approximation to develop an method of order four for the solution of a wave equation in polar co-ordinates with a significant first-order derivative term. In this paper, using nine grid points, we discuss a new three-level implicit spline in the tension finite difference method of accuracy two in time and four in space for the solution of a one-space dimensional second-order quasilinear hyperbolic equation. In Section , we propose a new three-level Numerov-type finite difference method based on spline in tension approximation.

Results

Conclusion