Stabilized Runge-Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized Runge-Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when stiffness is only induced by a few components, as in the presence of spatially local mesh refinement, their efficiency deteriorates. To remove the crippling effect of a few severely stiff components on the entire system of differential equations, we derive a modified equation, whose stiffness solely depend on the remaining mildly stiff components. By applying stabilized Runge-Kutta methods to this modified equation, we then devise an explicit multirate Runge-Kutta-Chebyshev (mRKC) method whose stability conditions are independent of a few severely stiff components. Stability of the mRKC method is proved for a model problem, whereas its efficiency and usefulness are demonstrated through a series of numerical experiments.

翻译：稳定的龙格-库塔方法对于大型硬性非线性差分方程式系统的数字解决方案特别有效,因为它们是完全明确的。例如,对于半分立的抛物线问题,稳定的龙格-库塔方法克服了标准方法的严格稳定性条件,而没有牺牲明确性。然而,当硬性方法仅由少数组成部分引起时,如在空间上局部网格改进时,其效率下降。为了消除少数严重僵硬的成分对整个差异方方方程式系统的瘫痪效应,我们得出一个经过修改的方程式,其僵硬性完全取决于其余的轻度硬性成分。通过对这个经过修改的方程式采用稳定的龙格-库塔方法,我们随后设计了一种明确的多率龙格-Kutta-Chebyshev(mRKC)方法,其稳定性条件独立于少数极为严格的组成部分。MRKC方法的稳定被证明是一个模型问题,而其效率和有用性则通过一系列数字实验得到证明。