Multirate schemes for conservation laws or convection-dominated problems seem to come in two flavors: schemes that are locally inconsistent, and schemes that lack mass-conservation. In this paper these two defects are discussed for one-dimensional conservation laws. Particular attention will be given to monotonicity properties of the multirate schemes, such as maximum principles and the total variation diminishing (TVD) property. The study of these properties will be done within the framework of partitioned Runge-Kutta methods.

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Modelling, Analysis and Simulation [MAS]
Multiscale Dynamics

Hundsdorfer, W., Mozartova, A., & Savcenco, V. (2007). Analysis of explicit multirate and partitioned Runge-Kutta schemes for conservation laws. Modelling, Analysis and Simulation [MAS]. CWI.