Implicit High Order Strong Stability Preserving Runge-Kutta Time Discretizations

Abstract

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this burden to Department of Defense, Washington Headquarters Services, Directorate for Information , 121 4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply wnn a collection ot intormanon it it does not display a currently valid OMB control number PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. REPORT DATE (DD-MM-YYYY) SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION / AVAILABILITY STATEMENTDistribution A: Approved for Public Release SUPPLEMENTARY NOTES ABSTRACTThis research involved the investigation, development, and testing of diagonally split RungeKutta (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. We investigated diagonally split Runge-Kutta (DSRK) methods to identify and test unconditionally strong stability preserving (SSP) methods, and implicit SSP timestopping methods to find methods with a large SSP coefficient. We found that DSRK methods which are unconditionally SSP reduce to first order for the stepsizes of interest, and introduced an analysis which explains this phenomenon and shows that it is unavoidable. We found optima; 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. We now have a complete analysis of implicit SSP RungeKutta methods and demonstrations of the need for the SSP property in solutions of hyperbolic PDEs with shocks. Summary of Aims and ResultsStrong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. SSP methods preserve the strong stability properties -in any norm, seminorm or convex functional -of the spatial discretization c...