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 partial differential equations (PDEs). The particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor 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 monitor function, and we present a monitor function that yields an optimal rate of convergence. Motivated by efficiency considerations for problems in two or more space dimensions, we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical PDE. The accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time-step control mechanism is integrated into the scheme.