Implicit High Order Strong Stability Preserving Runge-Kutta Time Discretizations

Abstract

Abstract : This research involved the investigation, development, and testing of diagonally split Runge- Kutta (DSRK) methods and implicit Runge-Kutta methods with reference to their strong stability preserving (SSP) properties for large time-steps. The research found that DSRK methods which are unconditionally SSP reduce to first order for the stepsizes of interest, and the PI introduced an analysis which explains this phenomenon and shows that it is unavoidable. The PI and her students developed a methodology for finding optimal implicit SSP Runge--Kutta methods up to order six (which is the maximal possible order for these methods) and eleven stages, and found that the effective SSP coefficient can be no more than two, making these methods not competitive with explicit methods for most applications, but useful in a carefully chosen subset of problems. The results of this grant are a complete analysis of implicit SSP Runge--Kutta methods and the SSP properties, which demonstrate the need for the SSP property in solutions of hyperbolic PDEs with shocks.