The Two-Dimensional Riemann Problem in Gas Dynamics: 1st Edition (Hardback) book cover

The Two-Dimensional Riemann Problem in Gas Dynamics

1st Edition

By Jiequan Li, Tong. Zhang, Shuli Yang

Chapman and Hall/CRC

312 pages

Purchasing Options:$ = USD
Hardback: 9780582244085
pub: 1998-08-21

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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.


"A complete and rigorous study…"

-Mathematical Reviews

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Table of Contents



Geometry of Characteristics and Discontinuities

Riemann Solution Geometry of Conservation Laws

Scalar Conservation Laws

One-Dimensional Scalar Conservation Laws

The Generalized Characteristic Analysis Method

The Four-Wave Riemann Problem

Mach-Reflection-Like Configuration of Solutions

Zero-Pressure Gas Dynamics

Characteristics and Bounded Discontinuities

Simultaneous Occurrence of Two Blowup Mechanisms

Delta-Shocks, Generalized Rankine-Hugoniot Relations and Entropy Conditions

The One-Dimensional Riemann Problem

The Two-Dimensional Riemann Problem

Riemann Solutions as the Limits of Solutions to Self-Similar Viscous Systems

Pressure-Gradient Equations of the Euler System

The Pme-Dimensional Riemann Problem

Characteristics, Discontinuities, Elementary Waves, and Classifications

The Existence of Solutions to a Transonic Pressure-Gradient Equation in an Elliptic Region with Degenerate Datum

The Two-Dimensional Riemann Problem and Numerical Solutions

The Compressible Euler Equations

The Concepts of Characteristics and Discontinuities

Planar Elementary Waves and Classification

PSI Approach to Irrotational Isentropic Flow

Analysis of Riemann Solutions and Numerical Results

Two-Dimensional Riemann Solutions with Axisymmetry


Author Index

About the Series

Monographs and Surveys in Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
SCIENCE / Mathematical Physics