To solve PDE problems with different time scales that are localized in space, multirate time stepping is examined. We introduce a self-adjusting multirate time stepping strategy, in which the step size at a particular grid point is determined by the local temporal variation of the solution, instead of using a minimal single step size for the whole spatial domain. The approach is based on the `method of lines', where first a spatial discretization is performed, together with local error estimates for the resulting semi-discret system. We will primarily consider implicit time stepping methods, suitable for parabolic problems. Our multirate strategy is tested on several parabolic problems in one spatial dimension (1D
