Implicit High Order Strong Stability Preserving Runge-Kutta Time Discretizations

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.

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Document Details

Document Type
Technical Report
Publication Date
Feb 05, 2009
Accession Number
ADA495116

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People

  • Sigal Gottlieb

Organizations

  • University of Massachusetts

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  • Air Force
  • Algorithms
  • Cauchy Problem
  • Coefficients
  • Differential Equations
  • Equations
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  • Euler Equations
  • Massachusetts
  • Mathematics
  • Partial Differential Equations
  • Polynomials
  • Runge Kutta Method
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  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)