An error analysis is presented for explicit partitioned Runge-Kutta methods and multirate methods applied to conservation laws. The interfaces, across which different methods or time steps are used, lead to order reduction of the schemes. Along with cell-based decompositions, also flux-based decompositions are studied. In the latter case mass conservation is guaranteed, but it will be seen that the accuracy may deteriorate.
Journal of Scientific Computing
Multiscale Dynamics

Hundsdorfer, W., Ketcheson, D. I., & Savostianov, I. (2015). Error Analysis of Explicit Partitioned Runge-Kutta Schemes for Conservation Laws. Journal of Scientific Computing, 63, 633–653. doi:10.1007/s10915-014-9906-1