Communications on Applied Mathematics and Computation >

2021 , Vol. 3 >Issue 4: 671 - 700

DOI: https://doi.org/10.1007/s42967-020-00098-y

ORIGINAL PAPER

Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

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  • 1 Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA;
    2 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

Received date: 2019-12-26

  Revised date: 2020-07-17

  Online published: 2021-11-25

Supported by

Research supported by NSF Grants DMS-1719410 and DMS-2010107, and by AFOSR Grant FA9550-20-1-0055.

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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

Cite this article

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 . DOI: 10.1007/s42967-020-00098-y

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