Systems of conservation laws. Two-dimensional Riemann problems.

*(English)*Zbl 0971.35002
Progress in Nonlinear Differential Equations and their Applications. 38. Boston, MA: Birkhäuser. xv, 317 p. (2001).

The main purpose of this book is to give an introduction to the Riemann problem for 2-D systems of conservation laws \(U_t+F(U)_x+G(U)_y=0\), \(U\in{\mathbb R}^m\). The first part is devoted to main results of the one-dimensional theory which are important for further presentation and include Lax’s solutions to the Riemann problem, Glimm’s scheme, and large-time asymptotics.

In the main part (second part), which consists of Chapters 5-11, the author develops the theory of a two-dimensional Riemann problem. Chapter 5 contains the solution to a four-constant Riemann problem for scalar equations. Chapter 6 addresses the issue of mixed type which occurs for self-similar solutions. In Chapter 7 the Euler system is studied. In particular, hurricane and tornado types of waves are found among axisymmetry solutions to the Riemann problem. Chapter 8 contains conjectured structures of solutions to the four-constant Riemann problem for the Euler system. Chapter 9 deals with the pressure-gradient system which is reduced to some system of mixed type. In Chapter 10 the pressureless system is treated. It is interesting that here unbounded solutions of Dirac’s delta type generally appear. The two-dimensional Burgers system is investigated in Chapter 11 including von Neumann paradox in the transition from regular reflection to Mach reflection. The last Chapter 12, which forms Part III, presents numerical schemes for two-dimensional conservation laws and includes positive schemes of Liu and Lax, as well as basic and popular schemes (the upwind schemes, Lax-Friedrichs scheme, Godunov method), and higher-order methods.

In the main part (second part), which consists of Chapters 5-11, the author develops the theory of a two-dimensional Riemann problem. Chapter 5 contains the solution to a four-constant Riemann problem for scalar equations. Chapter 6 addresses the issue of mixed type which occurs for self-similar solutions. In Chapter 7 the Euler system is studied. In particular, hurricane and tornado types of waves are found among axisymmetry solutions to the Riemann problem. Chapter 8 contains conjectured structures of solutions to the four-constant Riemann problem for the Euler system. Chapter 9 deals with the pressure-gradient system which is reduced to some system of mixed type. In Chapter 10 the pressureless system is treated. It is interesting that here unbounded solutions of Dirac’s delta type generally appear. The two-dimensional Burgers system is investigated in Chapter 11 including von Neumann paradox in the transition from regular reflection to Mach reflection. The last Chapter 12, which forms Part III, presents numerical schemes for two-dimensional conservation laws and includes positive schemes of Liu and Lax, as well as basic and popular schemes (the upwind schemes, Lax-Friedrichs scheme, Godunov method), and higher-order methods.

Reviewer: Evgeniy Panov (Novgorod)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35L65 | Hyperbolic conservation laws |

35A35 | Theoretical approximation in context of PDEs |