Implicit-Explicit Strong Stability Preserving Runge-Kuta Methods with High Linear Order

Abstract

High order strong stability preserving (SSP) time discretizations have been extensively used with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. Explicit SSP Runge--Kutta methods exist only up to fourth order, and implicit SSP Runge--Kutta methods exist only up to sixth order. When solving linear autonomous problems, the order conditions simplify and this order barrier is lifted: SSP Runge--Kutta methods of any linear order exist. In this work, we extend the concept of varying orders of accuracy for linear and non linear components to the class of implicit-explicit (IMEX) Runge--Kutta methods methods. We formulate an optimization problem for implicit-explicit (IMEX) SSP Runge--Kutta methods and find implicit methods with large linear stability regions that pair with known explicit SSP Runge--Kutta methods of orders plin = 3, 4, 6 as well as optimized IMEX SSP Runge--Kutta pairs that have high linear order and nonlinear orders p = 2, 3, 4. These methods are then tested on sample problems to verify order of convergence and to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems.