This article discusses an adaptive mesh moving technique that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of partial differential equations in two space dimensions and time. The mesh moving technique is based on an algebraic node movement function determined from the propagation of significant error regions. The algorithms is designed to be flexible, so that it can be used with many existing finite difference and finite element methods. To test the mesh moving algorithm, the authors implemented it in a system code with an initial mesh generator and a MacCormack finite volume scheme on quadralateral cells for hyperbolic vector systems. Results are presented for several computational examples. The moving mesh scheme reduces dispersion errors near shocks and wave fronts and thereby reduces the grid requirements necessary to compute accurate solutions while increasing computational efficiency.