General Framework for Achieving Textbook Multigrid Efficiency: One-dimensional Euler Example

James L. Thomas
Computational Modeling and Simulation Branch
Mail Stop 128
NASA Langley Research Center
Hampton, Virginia 23681

Boris Diskin
Institute for Computer Applications in Science and Engineering
Mail Stop 132C
NASA Langley Research Center
Hampton, Virginia 23681

Achi Brandt
The Weizmann Institute of Science
Rehovot 76100, Israel

Jerry C. South, Jr.
Williamsburg, Virginia 23185


Abstract

A general multigrid framework is discussed for obtaining textbook efficiency to solutions of the compressible Euler and Navier-Stokes equations in conservation law form. The general methodology relies on a distributed relaxation procedure to reduce errors in regular (smoothly varying) flow regions; separate and distinct treatments for each of the factors (elliptic and/or hyperbolic) are used to attain optimal reductions of errors. Near boundaries and discontinuities (shocks), additional local relaxations of the conservative equations are necessary . Example calculations are made for the quasi-one-dimensional Euler equations; the calculations illustrate the general procedure.

Key words. textbook multigrid efficiency, distributed relaxation, Euler equations

Subject classification. Applied and Numerical Mathematics