Strong Stability Preserving Integrating Factor Runge--Kutta Methods

Abstract

Strong stability preserving (SSP) Runge--Kutta methods are often desired when evolving in time problems that have two components that have very different time scales. Where the SSP property is needed, it has been shown that implicit and implicit-explicit methods have very restrictive time-steps and are therefore not efficient. For this reason, SSP integrating factor methods may offer an attractive alternative to traditional time-stepping methods for problems with a linear component that is stiff and a nonlinear component that is not. However, the strong stability properties of integrating factor Runge--Kutta methods have not been established. In this work we show that it is possible to define explicit integrating factor Runge--Kutta methods that preserve the desired strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping, or even given weaker conditions. We define sufficient conditions for explicit integrating factor Runge--Kutta methods to be SSP, namely, that they are based on explicit SSP Runge--Kutta methods with nondecreasing abscissas. We find such methods of up to fourth order and up to ten stages, analyze their SSP coefficients, and prove their optimality in a few cases. We test these methods to demonstrate their convergence and to show that the SSP time-step predicted by the theory is generally sharp and that the nondecreasing abscissa condition is needed in our test cases. Finally, we show that on typical total variation diminishing linear and nonlinear test cases our new explicit SSP integrating factor Runge--Kutta methods out-perform the corresponding explicit SSP Runge--Kutta methods, implicit-explicit SSP Runge--Kutta methods, and some well-known exponential time differencing methods.