Systems of Conservation Laws: Two-Dimensional Riemann Problems

ISBN-13: 9781461266310 / Angielski / Miękka / 2012 / 320 str.

Yuxi Zheng

Opis

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.

Spis treści

1 Problems.- 1.0 Outline.- 1.1 Some models.- 1.2 Basic problems.- 1.2.1 Probing problems.- 1.3 Some solutions.- 1.4 von Neumann paradoxes.- 1.5 End notes.- I Basics in One Dimension.- 2 One-dimensional Scalar Equations.- 2.1 The 1-D Burgers equation.- 2.2 Discontinuities and weak solutions.- 2.3 Rankine—Hugoniot relation.- 2.4 Nonuniqueness and entropy conditions.- 2.5 Some existence and uniqueness results.- 2.6 Some simple numerical schemes.- Exercises.- 3 Riemann Problems.- 3.1 The isentropic Euler system.- 3.1.1 Rarefaction waves.- 3.1.2 Discontinuous solutions.- 3.1.3 Entropy conditions.- 3.2 The adiabatic Euler system for polytropic gases.- 3.2.1 Rarefaction waves.- 3.2.2 Discontinuity.- 3.2.3 The entropy condition.- 3.2.4 Solutions.- 3.3 Lax’s Riemann solutions.- 3.3.1 Hyperbolicity and genuine nonlinearity.- 3.3.2 The Riemann problem.- 3.3.3 Continuous solutions.- 3.3.4 Discontinuous solutions.- 3.3.5 Lax’s entropy condition.- 3.3.6 Complete solutions.- 3.4 Nonconvex equations and viscous profiles.- 3.4.1 Nonconvex scalar equations.- 3.4.2 Viscous profiles.- 3.4.3 Stable viscous profiles.- 3.5 End notes and further references.- 4 Cauchy Problems.- 4.1 Smooth solutions.- 4.1.1 A new proof of blow-up in the scalar case.- 4.1.2 Systems of two equations and Riemann invariants.- 4.1.3 Blow-up and smooth solutions in systems of two equations.- 4.1.4 Remarks.- 4.2 Wave interactions.- 4.2.1 Scalar elementary wave interactions.- 4.2.2 The isentropic Euler system.- 4.3 Glimm’s scheme.- 4.3.1 Glimm’s scheme.- 4.3.2 Estimates.- 4.3.3 Compactness.- 4.3.4 Consistency.- 4.3.5 An example of single shocks.- 4.3.6 An example with large data (Nishida’s result).- 4.4 Generalized Riemann problems.- 4.4.1 Convex scalar equations.- 4.4.2 Nonconvex scalar equations.- 4.5 2.- 7.6.2 Inner-field equations for ? ? 2.- 7.6.3 Inner-field solutions for ? = 2.- 7.6.4 Inner-field solutions for 1 > ? > 2.- 7.6.5 The case ? = 1.- 7.7 Intermediate field solutions for u0 < 0.- 7.8 Rankine—Hugoniot relation.- 7.9 Shock wave solutions for u0 < 0.- 7.9.1 Shocks without swirls.- 7.9.2 General shock solutions.- 7.10 Summary.- 7.10.1 ?0=0 u0 ? 0, ? ? 1.- 7.10.2 ?0=0 u0 < 0, ? ? 1.- 7.10.3 ?0>0 u0 = 0, ? ? 1.- 7.10.3.A ? = 2.- 7.10.3.B ? > 2.- 7.10.3.C 1 < ? < 2.- 7.10.3.D ? = 1.- 7.10.4 ?0>0 u0 > 0, ? = 2.- 7.10.5 ?0>0 u0 > 0, ? > 2.- 7.10.6 ?0>0 u0 > 0, 1 < ? < 2.- 7.10.7 ?0>0 u0 > 0, ? = 1.- 7.10.8 ?0>0 u0 < 0, ? = 2.- 7.10.9 ?0>0 u0 2.- 7.10.10 ?0>0 u0 < 0, 1 < ? < 2.- 7.10.11 ?0>0 u0 < 0, ? = 1.- 7.10.12 Physical description of the solutions.- 7.11 End notes.- 7.12 Appendices.- 7.12.A Finiteness of the parameters at point (1, 0, 0).- 7.12.B Proof of Lemma 7.15.- 7.13 Exercises.- 8 Plausible Structures for 2-D Euler Systems.- 8.1 The four-wave Riemann problem.- 8.2 Planar elementary waves.- 8.3 Classification/reduction.- 8.4 Some plausible structures.- 8.5 Numerical experiments.- 8.6 Vortex sheets for the incompressible Euler system.- 9 The Pressure-Gradient Equations of the Euler Systems.- 9.1 A simple splitting example.- 9.2 The pressure-gradient system.- 9.3 A four-wave Riemann problem.- 9.4 An elliptic result.- 9.5 End notes.- 9.6 Appendix.- 10 The Convective Systems of the Euler Systems.- 10.1 Systems.- 10.2 Unbounded solutions and delta waves.- 10.3 1-D theory.- 10.4 2-D Riemann solutions.- 10.5 End notes.- 11 The Two-dimensional Burgers Equations.- 11.1 Small wedge angle asymptotics.- 11.2 Weak incident shock problem.- 11.3 Weak incident shock asymptotics.- 11.4 Core region asymptotic equations.- 11.5 Initial boundary values for the 2-D Burgers system.- 11.6 Numerical solutions.- 11.7 Theoretical approaches.- 11.7.1 Shock conditions and characteristics.- 11.7.2 Regular reflection.- 11.7.3 von Neumann paradox.- 11.7.4 Global transonic problems.- 11.7.5 Riemann problems.- 11.8 End notes.- Exercises.- III Numerical schemes.- 12 Numerical Approaches.- 12.1 Generalities.- 12.2 Upwind schemes.- 12.2.1 Intuitive schemes.- 12.2.2 Linear upwind schemes.- 12.2.3 Nonlinear upwind schemes.- Exercises.- 12.3 Lax—Friedrichs scheme.- 12.4 Godunov method.- 12.5 Approximate Riemann solver.- 12.6 Higher order methods.- 12.6.1 Lax—Wendroff scheme.- 12.6.2 Slope limiter.- 12.6.3 Flux limiter.- 12.6.4 TVD (total variation diminishing) fluxes.- 12.7 Positive schemes.- 12.7.1 Motivation.- 12.7.2 Nonnegative partition (positivity) principle.- 12.7.3 One-dimensional positive schemes.- 12.7.4 Multidimensional positive schemes.- 12.7.5 Symmetrizable positive schemes.- List of Symbols.