Enforcing strong stability of explicit Runge--Kutta methods with superviscosity

Abstract

A time discretization method is called strongly stable, if the norm of its numerical solution is nonincreasing. It is known that, even for linear semi-negative problems, many explicit Runge--Kutta (RK) methods fail to preserve this property. In this paper, we enforce strong stability by modifying the method with superviscosity, which is a numerical technique commonly used in spectral methods. We propose two approaches, the modified method and the filtering method for stabilization. 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. The bound we derived for general dispersive-diffusive superviscosity is also verified to be sharp numerically. For nonlinear problems, a filtering method is investigated for stabilization. Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximation of conservation laws are performed to validate our analysis and to test the performance.