Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity  

作　　者：Zheng Sun Chi-Wang Shu 

机构地区：[1]Department of Mathematics,The Ohio State University,Columbus,OH 43210,USA [2]Division of Applied Mathematics,Brown University,Providence,RI02912,USA

出　　处：《Communications on Applied Mathematics and Computation》2021年第4期671-700,共30页应用数学与计算数学学报（英文）

基　　金：supported by NSF Grants DMS-1719410 and DMS-2010107;by AFOSR Grant FA9550-20-1-0055.

摘　　要：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.

关 键 词：Runge-Kutta(RK)methods Strong stability Superviscosity Hyperbolic conservation laws Discontinuous Galerkin methods 

分 类 号：O17[理学—数学]