A fourth-order finite difference method based on spline in tension approximation for the solution of one-space dimension
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RESEARCH
Open Access
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 Mohanty1* and Venu Gopal2 *
Correspondence: [email protected]; [email protected] 1 Department of Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, 110021, India Full list of author information is available at the end of the article
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 Keywords: second-order quasilinear hyperbolic equation; spline in tension; wave equation in polar coordinates; stability analysis; maximum absolute errors
1 Introduction We consider the one-space dimensional second-order quasi-linear hyperbolic equation ∂ u ∂ u = A(x, t, u) + g(x, t, u, ux , ut ), ∂t ∂x
< x < , t >
(.)
with the following initial conditions: u(x, ) = a(x),
ut (x, ) = b(x),
≤x≤
(.)
t ≥ .
(.)
and the boundary conditions u(, t) = p (t),
u(, t) = p (t),
We assume that the conditions (.) and (.) are given with sufficient smoothness to maintain the order of accuracy in the numerical method under consideration. © 2013 Mohanty and Gopal; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Mohanty and Gopal Advances in Difference Equations 2013, 2013:70 http://www.advancesindifferenceequations.com/content/2013/1/70
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. Many efforts are going on to develop efficient and high accuracy methods for such types of partial differential equations. The term ‘spline’ in the spline function arises from the prefabricated wood or plastic curve board, which is called spline and is used by a draftman to plot smooth curves through connecting the known points. In early , the cubic spline method was proposed to be applied on the differential equation to get their numerical solutions, and it was in that Raggett et al. [] and Fleck, Jr. [] su
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