论文标题
刚性稳定的多重方法的明确稳定方法
Explicit stabilized multirate method for stiff differential equations
论文作者
论文摘要
稳定的runge-kutta方法对于刚性非线性微分方程的大型系统的数值解决方案特别有效，因为它们是完全显式的。例如，对于半差异抛物线问题，稳定的runge-kutta方法克服了标准方法的严格稳定性条件，而不会牺牲明确性。但是，当刚度仅由几个组件诱导时，就像在空间局部网状细化的情况下一样，它们的效率会恶化。为了消除整个微分方程系统上一些严重僵硬的组件的残废效果，我们得出了一个修改的方程，其刚度仅取决于剩余的轻度刚性组件。通过将稳定的runge-kutta方法应用于此修改的方程式，我们然后设计了一个显式的多发性runge-kutta-kutta-chebyshev（MRKC）方法，其稳定性条件与一些严重僵硬的组件无关。 MRKC方法的稳定性证明了模型问题，而其效率和有用性是通过一系列数值实验证明的。
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.
