Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

doi:10.1007/s42967-020-00098-y

Abstract

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 modifed method and the fltering method. The modifed method is achieved by modifying the semi-negative operator with a high order superviscosity term; the fltering method is to post-process the solution by solving a difusive 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 difusive superviscosity for ensuring strong stability. For nonlinear problems, a fltering method is investigated. Numerical examples with linear non-normal ordinary diferential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.

Key words: Runge-Kutta (RK) methods, Strong stability, Superviscosity, Hyperbolic conservation laws, Discontinuous Galerkin methods

CLC Number:

Zheng Sun, Chi-Wang Shu. Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity[J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 671-700.

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