Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods With High Linear Order

摘要

High order strong stability preserving (SSP) time discretizations have proven beneficial for use with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. Implicit SSP Runge-Kutta methods exist only up to sixth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and this order barrier is lifted: implicit SSP Runge-Kutta methods of any linear order exist. In the current work we aim to find implicit SSP Runge-Kutta methods with large allowable time-step, that feature high linear order and simultaneously have varying designed orders of accuracy for nonlinear order. In this work we also 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, and present a method of this type.