A new unconditionally stable ADI compact scheme for the two-space-dimensional linear hyperbolic equation

Abstract

We formulate a new alternating direction implicit compact scheme of O(τ²+h ⁴) for the linear hyperbolic equation u _tt +2α u _t +β² u=u _xx +u _yy +f(x, y, t), 0<x, y<1, 0<t≤T, subject to appropriate initial and Dirichlet boundary conditions, where α>0 and β≥0 are real numbers. In this article, we show the method is unconditionally stable by the Von Neumann method. At last, numerical demonstrations are given to illustrate our result.

Keywords:

• linear hyperbolic equation
• damped wave equation
• telegraph equation
• unconditionally stable
• ADI compact difference scheme

2000 AMS Subject Classifications :

• 65M06
• 65M12
• 65M15

Acknowledgements

We wish to express our gratitude to the referees for their many valuable suggestions which improved this article. We would also like to thank Ou Hu, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, for his help in revising this paper. The work of J. Liu was supported by the National Natural Science Foundation of China (Tian-yuan Foundation) under grant 10626044. The work of K. Tang was supported by the National Natural Science Foundation of China under grant 60673060.