Computational Solution of Two-Dimensional Unsteady PDEs Using Moving Mesh Methods

Numerical experiments are described which illustrate some important features of the performance of moving mesh methods for solving two-dimensional partial differential equations (PDEs). Here we are concerned with algorithms based on moving mesh methods proposed by W. Huang and R. D. Russell [ SIAM J. Sci. Comput. 20, 998 (1999)]. We show that the accuracy of the computations is strongly influenced by the choice of monitor function, and we present a monitor function which yields a higher rate of convergence than those that are commonly used. In an earlier paper [G. Beckett, J. A. Mackenzie, A. Ramage, and D. M. Sloan, J. Comput. Phys. 167, 372 (2001)], we demonstrated a robust and efficient algorithm for problems in one space dimension in which the mesh equation is decoupled from the physical PDE and the time step is controlled automatically. The present work extends this algorithm to deal with problems in two space dimensions.