A two-level compact ADI method for solving second-order wave equations

Abstract

A two-level compact alternating direction implicit (ADI) method is constructed for solving second-order wave equations with periodic boundary conditions. By using an H ² discrete energy method, it is shown that the compact ADI method is unconditionally convergent in the maximum norm with the convergence order of 2 in time and 4 in space. Asymptotic expansion, only in even powers of the mesh parameters (time step and spacings), of the difference solution is achieved. Using the expansion of the solution, high-order approximations could be achieved by Richardson extrapolations. Numerical experiments are included to support the theoretical results and show the effectiveness of our method.

Keywords:

• second-order hyperbolic equation
• compact ADI scheme
• discrete energy method
• convergence and stability
• asymptotic expansion and Richardson extrapolation

2000 AMS Subject Classifications:

• 65M06
• 65M12
• 65M15

Acknowledgements

We express our thanks to two anonymous reviewers whose invaluable critical comments and suggestions helped to improve this article. We are also grateful to editors of the journal for their constant encouragement and help. This research is supported by National Science Foundation for Young Scientists of China (No. 11001271) and by National Science Foundation of China (No. 11271068).