A New High-Order Approximation for the Solution of Two-Space-Dimensional Quasilinear Hyperbolic Equations

R K Mohanty,Suruchi Singh

doi:10.1155/2011/420608

Abstract

we propose a new high-order approximation for the solution of two-space-dimensional quasilinear hyperbolic partial differential equation of the form , , , subject to appropriate initial and Dirichlet boundary conditions , where and are mesh sizes in time and space directions, respectively. We use only five evaluations of the function as compared to seven evaluations of the same function discussed by (Mohanty et al., 1996 and 2001). We describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Some examples and their numerical results are provided to justify the usefulness of the proposed method.

Highlights

We describe the derivation procedure in details and discuss how our formulation is able to handle the wave equation in polar coordinates
Further we assume that u x, y, t ∈ C6, A x, y, t, u, B x, y, t, u ∈ C4, and φ x, y and ψ x, y are sufficiently differentiable function of as higher-order as possible
Second-order quasilinear hyperbolic partial differential equation with appropriate initial and boundary conditions serves as models in many branches of physics and technology

Summary

Introduction

We consider the following two-space dimensional quasilinear hyperbolic partial differential equation: utt A x, y, t, u uxx B x, y, t, u uyy g x, y, t, u, ux, uy, ut , 0 < x, y < 1, t > 0 1.1 subject to the initial conditions u x, y, 0 φ x, y , ut x, y, 0 ψ x, y , 0 ≤ x, y ≤ 1, Advances in Mathematical Physics and the boundary conditions u 0, y, t a0 y, t , u 1, y, t a1 y, t , 0 ≤ y ≤ 1, t ≥ 0, 1.3a u x, 0, t b0 x, t , u x, 1, t b1 x, t , 0 ≤ x ≤ 1, t ≥ 0, 1.3b where 1.1 is assumed to satisfy the hyperbolicity condition A x, y, t, u > 0 and B x, y, t, u > 0 in the solution region Ω ≡ { x, y, t : 0 < x, y < 1, t > 0}. Mohanty and Singh 17 have derived a high accuracy numerical method based on Numerov type discretization for the solution of one space dimensional nonlinear hyperbolic equations, in which they have shown that the linear scheme is unconditionally stable. In this paper, using nineteen grid-points, we derive a new compact three-level implicit numerical method of accuracy two in time and four in space for the solution of twospace dimensional quasilinear hyperbolic equation 1.1. In this method we require only five evaluations of the function g as compared to seven evaluations of the same function discussed in 6, 7.

Formulation of the Numerical Method

Derivation Procedure of the Approximation

Stability Analysis

Numerical Illustrations

Concluding Remarks