Moving mesh methods for Boussinesq equation

Abstract

The Boussinesq equation is a challenging problem both analytically and numerically. Owing to the complex dynamic development of small scales and the rapid loss of solution regularity, the Boussinesq equation pushes any numerical strategy to the limit. With uniform meshes, the amount of computational time is too large to enable us to obtain useful numerical approximations. Therefore, developing effective and robust moving mesh methods for these problems becomes necessary. In this work, we develop an efficient moving mesh algorithm for solving the two-dimensional Boussinesq equation. Our moving mesh algorithm is an extension of Tang and Tang (SIAM J. Numer. Anal. 2003; 41:487-515) for hyperbolic conservation laws and Zhang and Tang (Commun. Pure Appl. Anal. 2002; 1:57-73) for convection-dominated equations. Several numerical fluxes (Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (2nd edn). Springer: Berlin, 1999; WASCOM 99": 10th Conference on Waves and Stability in Continuous Media, Porto Ercole, Italy, 1999; 257-264; High-order Methods for Computational Physics. Springer: Berlin, 1999; 439-582; J. Sci. Comput. 1990; 5:127-149; SIAM J. Numer. Anal. 2003; 41:487-515; Commun. Pure Appl. Anal. 2002; 1:57-73) are also discussed. Numerical results demonstrate the advantage of our moving mesh method in resolving the small structures. Copyright © 2009 John Wiley & Sons, Ltd.