Pure and Applied Mathematics Quarterly
Volume 14 (2018)
Special Issue: In Honor of Chi-Wang Shu
Guest Editors: Jian-Guo Liu and Yong-Tao Zhang
Downwinding for preserving strong stability in explicit integrating factor Runge–Kutta methods
Pages: 3 – 25
Strong stability preserving (SSP) Runge–Kutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear non-innerproduct stability properties to be satisfied. Unlike the case for $L_2$ linear stability, implicit methods do not significantly alleviate the time-step restriction when the SSP property is needed. For this reason, when handling problems with a linear component that is stiff and a nonlinear component that is not, SSP integrating factor Runge–Kutta methods may offer an attractive alternative to traditional time-stepping methods. The strong stability properties of integrating factor Runge–Kutta methods where the transformed problem is evolved with an explicit SSP Runge–Kutta method with non-decreasing abscissas was recently established. However, these methods typically have smaller SSP coefficients (and therefore a smaller allowable time-step) than the optimal SSP Runge–Kutta methods, which often have some decreasing abscissas. In this work, we consider the use of downwinded spatial operators to preserve the strong stability properties of integrating factor Runge–Kutta methods where the Runge–Kutta method has some decreasing abscissas. We present the SSP theory for this approach and present numerical evidence to show that such an approach is feasible and performs as expected. However, we also show that in some cases the integrating factor approach with explicit SSP Runge–Kutta methods with non-decreasing abscissas performs nearly as well, if not better, than with explicit SSP Runge–Kutta methods with downwinding. In conclusion, while the downwinding approach can be rigorously shown to guarantee the SSP property for a larger timestep, in practice using the integrating factor approach by including downwinding as needed with optimal explicit SSP Runge–Kutta methods does not necessarily provide significant benefit over using explicit SSP Runge–Kutta methods with non-decreasing abscissas.
This paper is in honor of Prof. Chi-Wang Shu’s sixtieth birthday His pioneering work on SSP methods and his observations on downwinding inspired this paper. We wish him many many more productive, happy, and healthy years to inspire many mathematicians.
Received 18 July 2018
Published 2 April 2019