On the numerical solution of one-dimensional PDEs using adaptive methods based on equidistribution

Numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one-dimensional part ial differential equations (PDEs). The particular method considered here is an adaptive finite difference method based on the equidistribution of a mo nitor function and it is one of the moving mesh methods proposed by W,Huang , Y. Ren, and R. D. Russell (1994, SIAM J. Numer Anal. 31 709). We show how the accuracy of the computations is strongly dependent on the choice of mo nitor function, and we present a monitor function that yields an optimal ra te of convergence. Motivated by efficiency considerations for problems in t wo or more space dimensions, we demonstrate a robust and efficient algorith m in which the mesh equations are uncoupled from the physical PDE. The accu racy and efficiency of the various formulations of the algorithm are consid ered and a novel automatic time-step control mechanism is integrated into t he scheme. (C) 2001 Academic Press.