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

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.

Unconditionally stable modified methods for the solution of two‐ and three‐dimensional telegraphic equation with Robin boundary conditions

“…The parameters introduced in the process of proving the stability are obtained to be independent of the grid sizes whereas parameters introduced in methods discussed in Refs. , and depend on the grid sizes. Various numerical experiments are carried out to show the efficiency and accuracy of the proposed methods.…”

“…Let p = k h > 0 be the grid ratio parameter. A Numerov type approximation [14] with accuracy of O(k 2 + k 2 h 2 + h 4 ) for the solution of (2.1.1) for l, m = 1(1)N − 1, 0 < j ≤ J may be written as…”

“…Let be the grid ratio parameter. A Numerov type approximation with accuracy of O ( k 2 + k 2 h 2 + h 4 ) for the solution of for l , m = 1(1) N − 1, 0 < j ≤ J may be written as where a = α 2 k 2 , b = β 2 k 2 , and δ t and μ t are central and averaging operators with respect to time direction, respectively. Similarly, operators with respect to spatial directions are defined.…”

“…The proposed methods are three level implicit difference methods which are based on Numerov type approximations proposed in Refs. and . We discuss in detail the stability of the proposed methods by matrix stability method for solving telegraphic equation with Neumann boundary conditions.…”

“…Hence, solving telegraphic equation is of major interest. In recent past, several techniques [3][4][5][6][7][8][9][10][11][12][13][14][15][16] are discovered to solve telegraphic equation in one, two, and three dimensions. Dehghan and Mohebbi [3] proposed an implicit collocation method for the solution of two-dimensional linear hyperbolic equation with Dirichlet boundary conditions.…”