Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

Abstract

A time discretization method is called strongly stable (or monotone), if the norm of its numerical solution is nonincreasing. Although this property is desirable in various of contexts, many explicit Runge-Kutta (RK) methods may fail to preserve it. In this paper, we enforce strong stability by modifying the method with superviscosity, which is a numerical technique commonly used in spectral methods. Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators. We propose two approaches for stabilization: the modified method and the filtering method. The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term; the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity. For linear problems, most explicit RK methods can be stabilized with either approach without accuracy degeneration. Furthermore, we prove a sharp bound (up to an equal sign) on diffusive superviscosity for ensuring strong stability. For nonlinear problems, a filtering method is investigated. Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.