Multirate finite step methods

Many physical phenomena contain different scales. These phenomena can be modeled using partial differential equations (PDEs). Often, these PDEs can be split additively in a fast and a slow part. We extend the multirate infinitesimal steps methods (MIS) to multirate finite step methods (MFS). Both methods resolve the fast scale with an auxiliary differential equation with a fixed slow part. The order conditions of the MIS are derived under the assumption of an exactly resolved fast scale. In contrast, the MFS methods take numerical error of the (numerical) fast , solution into account. We introduce the MFS methods and derive their order conditions for different fast scale integrators. Finally, we give some numerical experiments and compare their stability areas.

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