Systems of Conservation Laws : Two-Dimensional Riemann Problems

[see attached] This work should serve as an excellent text for graduate students and researchers working in the important area of partial differential equations with a focus on problems involving conservation laws. Written in a clear, accessible style, the book emphasizes more recent results that will prepare readers to meet modern challenges in the subject, that is, to carry out theoretical, numerical, and asymptotical analysis. Key features of this work include: * broad range of topics, from the classical treatment to recent results, dealing with solutions to the 2-D compressible Euler

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.