Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation. (English) Zbl 1114.65347

Gao, Feng; Chi, Chunmei

Appl. Math. Comput. 187, No. 2, 1272-1276 (2007).

Summary: A few explicit difference schemes are discussed for the numerical solution of the linear hyperbolic equation \(u_{tt} + 2\alpha u_{t} + \beta ^{2}u = u_{xx} + f(x, t), \alpha > 0, \beta > 0\), in the region \(\Omega = \{(x, t)\mid a < x < b, t > 0\}\) subject to appropriate initial and Dirichlet boundary conditions, where \(\alpha\) and \(\beta\) are real numbers. The proposed scheme is showed to be unconditionally stable, and a numerical result is presented.

MSC:

65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L15	Initial value problems for second-order hyperbolic equations

Keywords:

second-order linear hyperbolic equation; damped wave equation; telegraph equation; Padé approximation; stability; numerical result

References:

[1]	Twizell, E.H., An explicit difference method for the wave equation with extended stability range, Bit, 19, 378-383, (1979) · Zbl 0441.65066
[2]	Mohanty, R.K.; Jain, M.K.; George, K., On the use of high order difference methods for the system of one space second order non-linear hyperbolic equations with variable coefficients, I. comp. appl. math., 72, 421-431, (1996) · Zbl 0877.65066
[3]	Ciment, M.; Leventhal, S.H., A note on the operator compact implicit method for the wave equation, Math. comput., 32, 143-147, (1978) · Zbl 0373.35039
[4]	Mohanty, R.K.; Jam, M.K., An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation, Namer. meth. partial D. eq., 17, 684-688, (2001) · Zbl 0990.65101
[5]	Mohanty, R.K.; Jain, M.K.; Arora, U., An unconditionally stable AD1 method for the linear hyperbolic equation in three space dimensions, Int. J. comput. math., 79, 133-142, (2002) · Zbl 0995.65093
[6]	Mohanty, R.K., An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation, Appl. math. lett., 17, 101-105, (2004) · Zbl 1046.65076