An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients

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Abstract

We propose a three level implicit unconditionally stable difference scheme of O(k2 + h2) for the difference solution of second order linear hyperbolic equation utt + 2α(x, t)ut + β2(x, t)u = A(x, t)uxx + f(x, t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where A(x, t) > 0, α(x, t) > β(x, t)  0. The proposed formula is applicable to the problems having singularity at x = 0. The resulting tri-diagonal linear system of equations is solved by using Gauss-elimination method. Numerical examples are provided to illustrate the unconditionally stable character of the proposed method.

Introduction

We consider the one space dimensional linear second order hyperbolic equation with variable coefficients of the form2ut2+2α(x,t)ut+β2(x,t)u=A(x,t)2ux2+f(x,t)defined in the region Ω × [0 < t < T], where Ω = {x∣0 < x < 1} and T is a positive integer.

The initial conditions consist ofu(x,0)=ϕ(x),u(x,0)t=ψ(x),0x1,and boundary conditions consist ofu(0,t)=g0(t),u(1,t)=g1(t),t0,where u = u(x, t). We assume that A(x, t) > 0 in the region Ω × [0 < t < T]. For α, β, A are constants and α > β > 0, A = 1, the equation above represents a telegraph equation. We shall assume that the initial and boundary conditions are given with sufficient smoothness to maintain the order of accuracy of the difference scheme under consideration.

A rectilinear grid with sides parallel to coordinate axes is superimposed on Ω × [0 < t < T] with spacing h > 0 and k > 0 in x- and t-directions, respectively. λ = (k/h) > 0 is the mesh ratio parameter. The internal grid points (x, t) are given by x = xl = lh, 0  l  N + 1 with (N + 1)h = 1 and t = tj = jk, 0  j  M + 1 with (M + 1)k = T, where N and M are positive integers. Again let Ulj and ulj be the exact and approximate values of u at the grid point (l, j)  (xl, tj), respectively.

Difference schemes of O(k2 + h2) and O(k2 + h4) for the one-space dimensional linear hyperbolic equations with constant and variable coefficients were discussed in [1], [2], [3]. In most of the cases, these schemes are stable for 0 < λ < 1. Recently, Mohanty et al. [4], [5] have proposed three level implicit unconditionally stable difference schemes of O(k2 + h2) for the linear hyperbolic equations in two and three space dimensions, respectively. To the author’s knowledge no unconditionally stable difference scheme for the linear hyperbolic equation (1) is known in the literature so far. In this article, we describe a new three level implicit difference scheme (see Fig. 1) of O(k2 + h2) for the linear hyperbolic differential equation (1), which is stable for 0 < λ < ∞. The most important feature of our scheme is that we solve only tri-diagonal equations and that fictitious points are not needed at each time step along the boundary.

Section snippets

Mathematical details of stability analysis

Now, we discuss the finite difference method for the hyperbolic equation (1) and its stability analysis.

We need the following approximations for the partial derivatives of u. LetU¯ttlj=(Ulj+1-2Ulj+Ulj-1)/(k2)=Uttlj+O(k2),U¯tlj=(Ulj+1-Ulj-1)/(2k)=Utlj+O(k2),U¯xxlj=(Ul+1j-2Ulj+Ul-1j)/(h2)=Uxxlj+O(h2).

At the grid point (l, j), the proposed differential equation (1) may be discretized byU¯ttlj+2αljU¯tlj+(βlj)2Ulj=AljU¯xxlj+flj,where αlj=α(xl,tj), βlj=β(xl,tj), Alj=A(xl,tj), flj=f(xl,tj), etc.

Computational result

In order to test the efficiency and viability of the chosen method, we have solved the proposed differential equation (1) for different values of α(x, t) > 0, β(x, t) > 0 and A(x, t) > 0. The exact solution values for all cases are given by u = e−2t sinh x as the test procedure. The initial and boundary conditions and right hand side function f(x, t) can be obtained using the exact solution. The proposed scheme (11) is an implicit three level scheme. To start any computation, it is necessary to know the

Concluding remarks

In this article, we have discussed a new three level implicit difference scheme (11) of O(k2 + h2) for the difference solution of the linear hyperbolic equation (1). The scheme is stable for 0 < λ < ∞ and applicable to singular equations. We have also discussed an explicit scheme of O(k2) for the difference solution of u at t = k, which is also applicable to singular equations. The usefulness of the proposed method is exhibited from the numerical results. In all cases, we have chosen α(x, t) > β(x, t) > 0.

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