Second order explicit stabilized multirate method for stiff differential equations with error control

Explicit stabilized methods are highly efficient time integrators for large and stiff systems of ordinary differential equations especially when applied to semi-discrete parabolic problems. However, when local spatial mesh refinement is introduced, their efficiency decreases, since the stiffness is driven by only the smallest mesh element. A natural approach is to split the system into fast stiff and slower mildly stiff components. In this context, [A. Abdulle, M.J. Grote and G. Rosilho de Souza 2022] proposed the order one multirate explicit stabilized method (mRKC). We extend their approach to second order and introduce the new multirate ROCK2 method (mROCK2), which achieves high precision and allows a step-size strategy with error control. Numerical methods including the heat equation with local spatial mesh refinements confirm the accuracy and efficiency of the scheme.