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.

翻译：显式稳定方法对于大型刚性常微分方程组是高效的时间积分器，尤其适用于半离散抛物型问题。然而，当引入局部空间网格细化时，其效率会降低，因为刚度仅由最小网格单元主导。一种自然的方法是将系统分解为快速刚性分量和较慢的弱刚性分量。在此背景下，[A. Abdulle, M.J. Grote 和 G. Rosilho de Souza 2022] 提出了一阶多速率显式稳定方法 (mRKC)。我们将他们的方法扩展到二阶，并引入了新的多速率 ROCK2 方法 (mROCK2)，该方法实现了高精度，并允许采用具有误差控制的步长策略。包括具有局部空间网格细化的热方程在内的数值实验验证了该方案的准确性和效率。