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 onedimensional 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. It will also be seen that the incompatibility of consistency and mass-conservation holds for ‘genuine’ multirate schemes, but not for general partitioned methods.
R Ansorge , H. Bijl , A. Meister , T. Sonar
Notes on numerical fluid mechanics and multidiciplinary design
Multiscale Dynamics

Hundsdorfer, W, Mozartova, A, & Savcenco, V. (2013). Monotonicity Conditions for Multirate and Partitioned Explicit Runge-Kutta Methods,. In R Ansorge, H Bijl, A Meister, & T Sonar (Eds.), Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws (pp. 177–195). Springer.