The Two-Dimensional Riemann Problem in Gas Dynamics

by Yang; Shuli

• ISBN13: 9780582244085
• ISBN10: 0582244080
• Format: Hardcover
• Copyright: 1998-08-21
• Publisher: Chapman & Hall/

Summary

The Riemann problem is the most fundamental problem in the entire field of non-linear hyperbolic conservation laws. Since first posed and solved in 1860, great progress has been achieved in the one-dimensional case. However, the two-dimensional case is substantially different. Although research interest in it has lasted more than a century, it has yielded almost no analytical demonstration. It remains a great challenge for mathematicians.This volume presents work on the two-dimensional Riemann problem carried out over the last 20 years by a Chinese group. The authors explore four models: scalar conservation laws, compressible Euler equations, zero-pressure gas dynamics, and pressure-gradient equations. They use the method of generalized characteristic analysis plus numerical experiments to demonstrate the elementary field interaction patterns of shocks, rarefaction waves, and slip lines. They also discover a most interesting feature for zero-pressure gas dynamics: a new kind of elementary wave appearing in the interaction of slip lines-a weighted Dirac delta shock of the density function. The Two-Dimensional Riemann Problem in Gas Dynamics establishes the rigorous mathematical theory of delta-shocks and Mach reflection-like patterns for zero-pressure gas dynamics, clarifies the boundaries of interaction of elementary waves, demonstrates the interesting spatial interaction of slip lines, and proposes a series of open problems. With applications ranging from engineering to astrophysics, and as the first book to examine the two-dimensional Riemann problem, this volume will prove fascinating to mathematicians and hold great interest for physicists and engineers.

Table of Contents

Preface	vii
Preliminaries	1	(10)
Geometry of characteristics and discontinuities	1	(6)
Riemann solution geometry of conservation laws	7	(4)
Scalar conservation laws	11	(35)
One-dimensional scalar conservation laws	11	(5)
The generalized characteristic analysis method	16	(13)
The four-wave Riemann problem	29	(12)
Mach-reflection-like configuration of solutions	41	(5)
Zero-pressure gas dynamics	46	(111)
Characteristics and bounded discontinuities	49	(6)
Simultaneous occurrence of two blowup mechanisms	55	(3)
Delta-shocks, generalized Rankine-Hugoniot relations and entropy conditions	58	(7)
The one-dimensional Riemann problem	65	(8)
The two-dimensional Riemann problem	73	(46)
Riemann solutions as the limits of solutions to self similar viscous systems	119	(38)
Pressure-gradient equations of the Euler system	157	(57)
The one-dimensional Riemann problem	159	(5)
Characteristics, discontinuities, elementary waves and classifications	164	(15)
The existence of solutions to a transonic pressure-gradient equation in an elliptic region with degenerate datum	179	(9)
The two-dimensional Riemann problem and numerical solutions	188	(26)
The compressible Euler equations	214	(75)
The concepts of characteristics and discontinuities	215	(4)
Planar elementary waves and classification	219	(15)
PSI approach to irrotational isentropic flow	234	(3)
Analysis of Riemann solutions and numerical results	237	(39)
Two-dimensional Riemann solutions with axisymmetry	276	(13)
References	289	(10)
Author index	299