Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

• 作 者: Zheng Sun Chi-Wang Shu
• 机 构: Department of Mathematics,The Ohio State University,Columbus,OH 43210,USA;Division of Applied Mathem
• 出 处: 应用数学与计算数学学报
• 发表日期: 2021年4期
• 关键词: Runge-Kutta(RK)methods Strong stability Superviscosity Hyperbolic conservation l

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 con-texts,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 numeri-cal 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 degenera-tion.Furthermore,we prove a sharp bound(up to an equal sign)on diffusive superviscos-ity 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.