We present the piecewise parabolic method, a higher-order extension of Godunov's method. There are several new features of this method which distinguish it from other higher-order Godunov-type methods. We use a higher-order spatial interpolation than previously used, which allows for a steeper representation of discontinuities, particularly contact discontinuities. We introduce a simpler and more robust algorithm for calculating the nonlinear wave interactions used to compute fluxes. Finally, we recognize the need for additional dissipation in any higher-order Godunov method of this type, and introduce it in such a way so as not to degrade the quality of the results.
Bibliographical noteFunding Information:
I Work performed under the auspices of the Director, Office of Energy Research, Offrice of Basic Energy Sciences, Engineering, Mathematical, and Geosciences Division of the U.S. Department of Energy at the Lawrence Berkeley Laboratory under Contract DE-AC03-76SFOOO98, and at the Lawrence Livermore National Laboratory under Contract W-7405.ENG-48.
This work was performedu nderthe auspiceso f the Director, Oflice of Energy Research,O ftice of Basic Energy Sciences,E ngineeringM, athematicala, nd GeosciencesD ivision of the U.S. Departmenot f Energy at the Lawrence Berkeley Laboratory under Contract DE-AC03-76SFOOO98,a nd at the LawrenceL ivermoreN ational Laboratory underC ontract W-7405-ENG-48.