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In recent years, several sophisticated packages based on the method of lines (MOL) have been developed for the automatic numerical integration of time-dependent problems in partial differential equations (PDEs), notably for problems in one space dimension. These packages greatly benefit from the very successful developments of automatic stiff ordinary differential equation solvers. However, from the PDE point of view, they integrate only in a semiautomatic way in the sense that they automatically adjust the time step sizes, but use just .a fixed space grid, chosen a priori, for the entire calculation. For solutions possessing sharp spatial transitions that move, e.g., travelling wave fronts or emerging boundary and interior layers, a grid held fixed for the entire calculation is computationally inefficient, since for a good solution this grid often must contain a very large number of nodes. In such cases methods which attempt automatically to adjust the sizes of both the space and the time steps are likely to be more successful in efficiently resolving critical regions of high spatial and temporal activity. Methods and codes that operate this way belong to the realm of adaptive or moving-grid methods. Following the MOL approach, this paper is devoted to an evaluation and comparison, mainly based on extensive numerical tests, of three moving-grid methods for ID problems, viz., the finite-element method of Miller and co-workers, the method published by Petzold, and a method based on ideas adopted from Dorfi and Drury. Our examination of these three methods is aimed at assessing which is the most suitable from the point of view of retaining the acknowledged features of reliability, robustness, and efficiency of the conventional MOL approach. Therefore, considerable attention is paid to the temporal performance of the methods.

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