Adaptive mesh methods are valuable tools in improving the accuracy and efficiency of the numerical solution of evolutionary systems of partial differential equations. If the mesh moves to track fronts and large gradients in the solution, then larger time steps can be taken than if it were to remain stationary. We derive explicit differential equations for moving the mesh so that the time variation of the solution at the mesh points is minimized. Moving the mesh based on this approach allows for larger time steps but does not guarantee that the solution is well resolved in space. We maintain spatial accuracy when there are new emerging layers or wave fronts by adaptively rezoning the mesh points to equidistribute an error estimate. When using a multistep integration method, the past solution values are also interpolated so that the same multistep method can be used after rezoning. The resulting algorithm has very few problem-dependent numerical parameters and is appropriate for a large class of one-dimensional partial differential equations. We illustrate the performance of the algorithm by examples and demonstrate that the proposed algorithm is efficient and accurate when compared with other adaptive mesh strategies.