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.
, Tong. Zhang
, Shuli Yang (Academia Sinica
Country of Publication:
Series: Monographs and Surveys in Pure and Applied Mathematics
21 August 1998
Professional and scholarly
Further / Higher Education
Preface Preliminaries 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 References Author Index
Li, Jiequan; Zhang, Tong.; Yang, Shuli
Reviews for The Two-Dimensional Riemann Problem in Gas Dynamics
A complete and rigorous study -Mathematical Reviews Promo Copy