Fourth-order compact difference and alternating direction implicit schemes for telegraph equations

https://doi.org/10.1016/j.cpc.2011.11.023Get rights and content

Abstract

In this paper, two- and three-level compact difference and alternating direction implicit schemes are presented for the numerical solutions of one- and two-dimensional linear telegraph equations and telegraph equations with nonlinear forcing term. The stability and error estimates are given. The convergence rates of the present schemes are of order O(τ2+h4). Numerical experiments on model problems show that the present schemes are of high accuracy.

Introduction

The telegraph equation is one of the important equations of mathematical physics with applications in many different fields such as transmission and propagation of electrical signals [1], [2], vibrational systems [3], random walk theory [4], and mechanical systems [5], etc. The heat diffusion and wave propagation equations are particular cases of the telegraph equation.

Recently, increasing attention has been paid to the development, analysis, and implementation of stable methods for the numerical solutions of second-order hyperbolic equations. There have been many numerical methods for hyperbolic equations, such as the finite difference, the finite element, and the collocation methods, etc. See [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and literatures are therein. Mohanty et al. [17], [18] developed some alternating direction implicit schemes for the two- and three-dimensional linear hyperbolic equations. Most of these schemes are second-order accurate in both space and time. Some conditionally stable fourth-order compact difference schemes for the wave and telegraph equations were introduced in [8], [9], [13]. An advantage of compact difference schemes is that the schemes use narrower stencils, i.e., fewer neighboring nodes, and have less truncation error comparing to typical finite difference schemes. Various versions of compact schemes have been successfully developed and analyzed (cf. [22], [23]) and references are therein. The alternating direction implicit (ADI) methods were first introduced by Douglas, Peaceman, and Rachford [24], [25], [26], [27], which have an advantage of no increase in dimension of the coefficient matrices corresponding matrix equations. There are many extensions and a great variety of applications of ADI based on the finite difference or the finite element methods [28], [29], [30], [31], [32], [33].

In this paper, we propose two- and three-level compact difference and ADI compact difference schemes with high accuracy for numerical solutions of the one- and two-dimensional telegraph equations with Dirichlet boundary conditions. These compact schemes are unconditionally stable and have a truncation error of second- and fourth-order in time and space, respectively. The coefficient matrices of the matrix equations given by the schemes are symmetric and tridiagonal. The matrix equations can be effectively solved by using many linear solvers. Numerical experiments presented for the 1D and 2D telegraph equations show that the present schemes are of high accuracy. By comparing the numerical results obtained by the present methods with the ones of some other available methods, the present methods can be considered as practical and effective numerical techniques to solve telegraph equations.

In Section 2, we introduce the compact difference schemes for the 1D telegraph equation and consider the stability and error estimates of the schemes. In Section 3, we introduce the compact difference and ADI compact difference schemes for the 2D telegraph equation, and analyze the stabilities and error estimates of the schemes. Finally, we present the result of numerical experiments on test examples in Section 4.

Section snippets

One-dimensional telegraph equation

Let I=(0,X). Consider the 1D linear telegraph equation2ut2+cut+bup2ux2=f(x,t),(x,t)I×(0,T], with the initial and boundary conditionsu(x,0)=φ(x),ut(x,0)=ψ(x),u(0,t)=gl(t),u(X,t)=gr(t), where c, b, and p are constants with c, b0, and p>0, and the functions f, φ, and ψ are assumed to be sufficiently smooth.

Two-dimensional telegraph equation

Consider the 2D linear telegraph equation2ut2+cut+bup2ux2q2y2=f(x,y,t),(x,y)Q,t(0,T], with the initial and boundary conditionsu(x,y,0)=φ(x,y),ut(x,y,0)=ψ(x,y),u(x,y,t)=g(x,y,t),(x,y)Γ, where c, b, p, and q are constants with c, b0, p, q>0, and f, φ, ψ, and g are smooth functions, Q=(0,X)×(0,Y), and Γ is the boundary of Q.

The solution domain Ω={(x,y,t)|0xX,0yY,0tT} is discretized into grids described by the set {(xi,yj,tn)} of nodes, in which xi=ihx, yj=jhy, i=0,1,,Jx, j=0,

Numerical experiments

In this section, we present the result of numerical experiments to test the accuracy of the present schemes. The accuracy and convergence rates are measured by using the discrete L2- and L-norms defined in Sections 2.2 Stability and error estimates, 3.3 Stability and error estimates.

Conclusion and discussion

We proposed the two- and three-level compact difference and ADI compact difference schemes with high accuracy for the 1D and 2D telegraph equations with Dirichlet boundary conditions. The two-level schemes are unconditionally stable and the three-level compact difference and ADI compact difference schemes are unconditionally stable for parameters θ in the range of [1/4,1/2]. The proposed schemes are second- and fourth-order accurate in time and space, respectively. For the telegraph equations

Acknowledgements

The authors wish to express their sincere gratitude to the referees for their valuable comments and suggestions on this paper.

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    1

    The research of this author is partially supported by NNSF of China grants 10971204.

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    The research of this author was supported by Changwon National University in 2009.

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