Advances in Mathematical Physics
Volume 2011 (2011), Article ID 420608, 22 pages
http://dx.doi.org/10.1155/2011/420608
Research Article

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

R. K. Mohanty1 and Suruchi Singh1,2

1Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110 007, India
2Department of Mathematics, Aditi Mahavidyalaya, University of Delhi, Delhi 110 039, India

Received 31 March 2011; Revised 13 June 2011; Accepted 5 July 2011

Academic Editor: Ricardo Weder

Copyright © 2011 R. K. Mohanty and Suruchi Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

we propose a new high-order approximation for the solution of two-space-dimensional quasilinear hyperbolic partial differential equation of the form 𝑢 𝑡 𝑡 = 𝐴 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) 𝑢 𝑥 𝑥 + 𝐵 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) 𝑢 𝑦 𝑦 + 𝑔 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 , 𝑢 𝑥 , 𝑢 𝑦 , 𝑢 𝑡 ) , 0 < 𝑥 , 𝑦 < 1 , 𝑡 > 0 subject to appropriate initial and Dirichlet boundary conditions , where 𝑘 > 0 and ℎ > 0 are mesh sizes in time and space directions, respectively. We use only five evaluations of the function 𝑔 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.