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This work is based on the lecture notes of the course M742: Topics in Partial Dif ferential Equations, which I taught in the Spring semester of 1997 at Indiana Univer sity. My main intention in this course was to give a concise introduction to solving two-dimensional compressibleEuler equations with Riemann data, which are special Cauchy data. This book covers new theoretical developments in the field over the past decade or so. Necessary knowledge of one-dimensional Riemann problems is reviewed and some popularnumerical schemes are presented. Multi-dimensional conservation laws are more physical and the time has come to study them. The theory onbasicone-dimensional conservation laws isfairly complete providing solid foundation for multi-dimensional problems. The rich theory on ellip tic and parabolic partial differential equations has great potential in applications to multi-dimensional conservation laws. And faster computers make itpossible to reveal numerically more details for theoretical pursuitin multi-dimensional problems. Overview and highlights Chapter 1is an overview ofthe issues that concern us inthisbook. It lists theEulersystemandrelatedmodelssuch as theunsteady transonic small disturbance, pressure-gradient, and pressureless systems. Itdescribes Mach re flection and the von Neumann paradox. In Chapters 2-4, which form Part I of the book, we briefly present the theory of one-dimensional conservation laws, which in cludes solutions to the Riemann problems for the Euler system and general strictly hyperbolic and genuinely nonlinearsystems, Glimm's scheme, and large-time asymp toties.
List of contents
1 Problems.- 1.0 Outline.- 1.1 Some models.- 1.2 Basic problems.- 1.3 Some solutions.- 1.4 von Neumann paradoxes.- 1.5 End notes.- I Basics in One Dimension.- 2 One-dimensional Scalar Equations.- 3 Riemann Problems.- 4 Cauchy Problems.- II Two Dimensional Theory.- 5 A 2-D Scalar Riemann Problem.- 6 The 2-D Riemann problem and Pseudo-Characteristics.- 7 Axisymmetric and Self-similar Solutions.- 8 Plausible Structures for 2-D Euler Systems.- 9 The Pressure-Gradient Equations of the Euler Systems.- 10 The Convective Systems of the Euler Systems.- 11 The Two-dimensional Burgers Equations.- III Numerical schemes.- 12 Numerical Approaches.- List of Symbols.
