Explicit stabilized multirate method for stiff differential equations

TL;DR

This paper introduces an explicit multirate stabilized Runge-Kutta method that effectively handles systems with mixed stiffness levels, improving efficiency by isolating severely stiff components.

Contribution

The authors develop a modified equation approach enabling stabilized Runge-Kutta methods to efficiently solve systems with localized stiffness, leading to the mRKC multirate method.

Findings

Stability of the mRKC method is proven for a model problem.

Numerical experiments demonstrate improved efficiency.

Method effectively isolates severely stiff components.

Abstract

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.