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THEORETICAL AND PRACTICAL ASPECTS OF SOME INITIAL-BOUNDARY VALUE PROBLEMS IN FLUID DYNAMICS

IP.com Disclosure Number: IPCOM000128294D
Original Publication Date: 1976-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 25 page(s) / 76K

Publishing Venue

Software Patent Institute

Related People

Joseph Oliger: AUTHOR [+4]

Abstract

Initial-boundary value problems for several systems of partial differential equations from fluid dynamics are discussed. Both rigid wall and open boundary problems are treated. Boundary conditions are formulated and shown to yield well-posed problems for the Eulerian equations for gas dynamics~ the shallow-water equations, and linearized constant coefficient versions of the incompressible, anelastic equations. The "primitive" hydrostatic meteorological equations are shown to be ill-posed with any specification of local, pointwise boundary conditions. Analysis of simplified versions of this system illustrates the mechanism responsible for ill-posedness.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

THEORETICAL AND PRACTICAL ASPECTS OF SOME INITIAL- BOUNDARY VALUE PROBLEMS IN FLUID DYNAMICS

By Joseph Oliger and Arne Sundstrom

ABSTRACT

Initial-boundary value problems for several systems of partial differential equations from fluid dynamics are discussed. Both rigid wall and open boundary problems are treated. Boundary conditions are formulated and shown to yield well-posed problems for the Eulerian equations for gas dynamics~ the shallow-water equations, and linearized constant coefficient versions of the incompressible, anelastic equations. The "primitive" hydrostatic meteorological equations are shown to be ill-posed with any specification of local, pointwise boundary conditions. Analysis of simplified versions of this system illustrates the mechanism responsible for ill-posedness.

Introduction

There is now considerable interest in initial -boundary value problems for various systems of partial differential equations arising in fluid dynamics. This interest stems, primarily, from efforts to create useful computational models of various processes for the purposes of prediction (atmospheric processes, ocean circulation, etc.)'and the detailed study of various phenomena (convection~ flow in wind tunnels, lee waves, eddies, etc.). Such calculations are not new. As these computational models have become more accurate difficulties with the boundary conditions have become more evident. This has led first to the examination of the various discretizations used and then back to the differential equations whose approximate solutions are sought.

Such a backward sequence of events may seem surprising. Naturally, the initial-boundary-value problems for the differential equations should have been carefully examined first since we cannot expect our approximations to be reasonable if they approximate a problem which does not have reasonable solutions. The reason it has gone this way is clear.

1

2 t is natural tofirst examine the evidence where it appears and, as usual, the computations have been ahead of the analysis. The initial-boundary value problems for these systems of differential equations are not easy to analyze; and, in fact, adequate tools for a rather complete analysis have only recently become available stemming from the work of Kreiss [12,131

1 Computer Science Department, Stanford University, Stanford, California 94305. Supported in part by the Office of Naval Research under contract N00014-75-C-1132 and -the National Science Foundation under grant DCR75- 13497-

2 National Defence Research Institute, 104 50 Stockholm 80, Sweden.

Stanford University Page 1 Dec 31, 1976

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THEORETICAL AND PRACTICAL ASPECTS OF SOME INITIAL-BOUNDARY VALUE PROBLEMS IN FLUID DYNAMICS

The current interest has resulted in several works based on the classical energy method (e.g., Elvius and SundstrUm [91, Davies 15,61, de Rivas [7 1 and Dutton [81), which follow the earlier work...