THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures,and the solutions admit the concentration of mass.It is found that under the requirement of satisfying the over-compressing entropy condition:(i)there is a unique delta shock solution,corresponding to the case that has two strong classical Lax shocks;(ii)for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave,or two shocks with one being weak,there are infinitely many solutions,each consists of a delta shock and a rarefaction wave;(iii)there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves.These solutions are self-similar.Furthermore,for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data,there always exists a unique delta shock for at least a short time.It could be prolonged to a global solution.Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass(particle).Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified.This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases,that is strictly hyperbolic,and whose characteristics are both genuinely nonlinear.We also discuss possible physical interpretations and applications of these new solutions.

THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE NON-ISENTROPIC IMPROVED AW-RASCLE-ZHANG MODEL

In this paper,we study the Radon measure initial value problem for the nonisentropic improved Aw-Rascle-Zhang model.For arbitrary convex F(u)in this model we construct the Riemann solutions by elementary waves andδ-shock waves using the method of generalized characteristic analysis.We obtain the solutions constructively for initial data containing the Dirac measure by taking the limit of the solutions for that with three piecewise constants.Moreover,we analyze different kinds of wave interactions,including the interactions of theδ-shock waves with elementary waves.

The Perturbed Riemann Problem for a Geometrical Optics System

The perturbed Riemann problem for a hyperbolic system of conservation laws arising in geometrical optics with three constant initial states is solved.By studying the interactions among of the delta-shock,vacuum,and contact discontinuity,fourteen kinds of structures of Riemann solutions are obtained.The compound wave solutions consisting of delta-shocks,vacuums,and contact discontinuities are found.The single and double closed vacuum cavitations develop in solutions.Furthermore,it is shown that the solutions of the Riemann problem for the geometrical optics system are stable under certain perturbation of the initial data.Finally,the numerical results completely coinciding with theoretical analysis are presented.

Two-Dimensional Riemann Problems:Transonic Shock Waves and Free Boundary Problems

We are concerned with global solutions of multidimensional(M-D)Riemann problems for nonlinear hyperbolic systems of conservation laws,focusing on their global configurations and structures.We present some recent developments in the rigorous analysis of two-dimensional(2-D)Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations.In particular,we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.

An Approximate Riemann Solver for Advection-Diffusion Based on the Generalized Riemann Problem

We construct an approximate Riemann solver for scalar advection-diffusion equations with piecewise polynomial initial data.The objective is to handle advection and diffusion simultaneously to reduce the inherent numerical diffusion produced by the usual advection flux calculations.The approximate solution is based on the weak formulation of the Riemann problem and is solved within a space-time discontinuous Galerkin approach with two subregions.The novel generalized Riemann solver produces piecewise polynomial solutions of the Riemann problem.In conjunction with a recovery polynomial,the Riemann solver is then applied to define the numerical flux within a finite volume method.Numerical results for a piecewise linear and a piecewise parabolic approximation are shown.These results indicate a reduction in numerical dissipation compared with the conventional separated flux calculation of advection and diffusion.Also,it is shown that using the proposed solver only in the vicinity of discontinuities gives way to an accurate and efficient finite volume scheme.

THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE ONE-DIMENSIONAL CHAPLYGIN GAS EQUATIONS

In this article, we study the Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations. Under the generalized Rankine-Hugoniot conditions and the entropy condition, we constructively obtain the global existence of generalized solutions that explicitly exhibit four kinds of different structures. Moreover, we obtain the stability of generalized solutions by making use of the perturbation of the initial data.

MULTI-DIMENSIONAL RIEMANN PROBLEM OF SCALAR CONSERVATION LAW

This paper considers multi-dimensional Riemann problem in another kind of view. The author gets solution of (1.1)(1.2) in Theorem 3.4 and proves itu uniqueness. A new method of solution constructing is applied, which is different from the usual self-similar transformation. The author also discusses some generalized concepts in multi-dimensional situation (such as 'convex condition', 'left value' and 'right value', etc). An example is finally given to demonstrate that rarefaction wave solution of (1.1)(1.2) is not self-similar.

TWO-DIMENSIONAL RIEMANN PROBLEMS:FROM SCALAR CONSERVATION LAWS TO COMPRESSIBLE EULER EQUATIONS

In this paper we survey the authors＇ and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections： 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.

Guckenheimer structure of solution of Riemann problem with four pieces of constants in two space dimensions for scalar conservation laws

By using the generalized characteristic analysis method, the two-dimensional four-wave Riemann problem for scalar conservation laws, which is nonconvex along the y direction, was studied. Riemann solutions, which involve the Guckenheimer structure, were constructed.

Riemann problem of a Chapman-Jouguet combustion model for pressure-gradient system

In this paper,the Riemann problem of a Chapman-Jouguet combustion model for the pressure-gradient equations is considered.By analyzing in phase space,existence and uniqueness of the solution to the Riemann problem are obtained constructively under the global entropy conditions.

Numerical method of the Riemann problem for two-dimensional multi-fluid flows with general equation of state

Based on the classical Roe method, we develop an interface capture method according to the general equation of state, and extend the single-fluid Roe method to the two-dimensional （2D） multi-fluid flows, as well as construct the continuous Roe matrix for the whole flow field. The interface capture equations and fluid dynamic conservative equations are coupled together and solved by using any high-resolution schemes that usually suit for the single-fluid flows. Some numerical examples are given to illustrate the solution of 1D and 2D multi-fluid Riemann problems.

Generalized Riemann problem for gas dynamic combustion

The generalized Riemann problem for gas dynamic combustion in a neighborhood of the origin t 0 in the （x, t） plane is considered. Under the modified entropy conditions, the unique solutions are constructed, which are the limits of the selfsimilar Zeldovich-von Neumann-Dring （ZND） combustion model. The results show that, for some cases, there are intrinsical differences between the structures of the perturbed Riemann solutions and the corresponding Riemann solutions. Especially, a strong detonation in the corresponding Riemann solution may be transformed into a weak deflagration coalescing with the pre-compression shock wave after perturbation. Moreover, in some cases, although no combustion wave exists in the corresponding Riemann solution, the combustion wave may occur after perturbation, which shows the instability of the unburnt gases.

Nonlinear Riemann Problem for Nonlinear Elliptic Systems in Sobolev Space W_(1,p)(D)

The nonlinear Riemann problems were converted into nonlinear singular integral equ ations and the existence of the solution for the problem was proved by means of contract principle.

ADER Methods for Hyperbolic Equations with a Time-Reconstruction Solver for the Generalized Riemann Problem: the Scalar Case

The ADER approach to solve hyperbolic equations to very high order of accuracy has seen explosive developments in the last few years,including both methodological aspects as well as very ambitious applications.In spite of methodological progress,the issues of efficiency and ease of implementation of the solution of the associated generalized Riemann problem(GRP)remain the centre of attention in the ADER approach.In the original formulation of ADER schemes,the proposed solution procedure for the GRP was based on(i)Taylor series expansion of the solution in time right at the element interface,(ii)subsequent application of the Cauchy-Kowalewskaya procedure to convert time derivatives to functionals of space derivatives,and(iii)solution of classical Riemann problems for high-order spatial derivatives to complete the Taylor series expansion.For realistic problems the Cauchy-Kowalewskaya procedure requires the use of symbolic manipulators and being rather cumbersome its replacement or simplification is highly desirable.In this paper we propose a new class of solvers for the GRP that avoid the Cauchy-Kowalewskaya procedure and result in simpler ADER schemes.This is achieved by exploiting the history of the numerical solution that makes it possible to devise a time-reconstruction procedure at the element interface.Still relying on a time Taylor series expansion of the solution at the interface,the time derivatives are then easily calculated from the time-reconstruction polynomial.The resulting schemes are called ADER-TR.A thorough study of the linear stability properties of the linear version of the schemes is carried out using the von Neumann method,thus deducing linear stability regions.Also,via careful numerical experiments,we deduce stability regions for the corresponding non-linear schemes.Numerical examples using the present simplified schemes of fifth and seventh order of accuracy in space and time show that these compare favourably with conventional ADER methods.This paper is restricted to the one-dimensional scalar case with source term,but preliminary results for the one-dimensional Euler equations indicate that the time-reconstruction approach offers significant advantages not only in terms of ease of implementation but also in terms of efficiency for the high-order range schemes.

The Riemann problem for nonlinear degenerate wave equations

This paper studies the Riemann problem for a system of nonlinear degenerate wave equations in elasticity. Since the stress function is neither convex nor concave, the shock condition is degenerate. By introducing a degenerate shock under the generalized shock condition, the global solutions are constructively obtained case by case.

THE RIEMANN PROBLEM FOR THE NONLINEAR DEGENERATE WAVE EQUATIONS

In this paper we consider the Riemann problem for the nonlinear degenerate wave equations. This problem has been studied by Sun and Sheng, however the so-called degenerate shock solutions did not satisfy the R-H condition. In the present paper, the Riemann solutions of twelve regions in the v - u plane are completely constructed by the Liu-entropy condition. Our Riemann solutions are very different to that one obtained by Sun and Sheng in some regions.

THE RIEMANN PROBLEM FOR A TWO-DIMENSIONAL HYPERBOLIC SYSTEM OF CONSERVATION LAWS WITH NON-CLASSICAL WEAK SOLUTIONS:SOME ONE-J AND NON-R INITIAL DATA

The Riemann problem for a two-dimensional 2 x 2 nonstrictly hyperbolic system of nonlinear conservation laws has been considered for constant initial data having discontinuities on three rays with vertex at the origin. The solutions are constructed for some one-J and non-R initial data. One kind of new discontinuity, which is labelled as the delta-shock wave, appears in some solutions. The delta-shock wave is a discontinuity plane that is the suport of a generalized function.

Riemann problem for one-dimensional binary gas enhanced coalbed methane process

With an extended Langmuir isotherm, a Riemann problem for one-dimensional binary gas enhanced coalbed methane (ECBM) process is investigated. A new analytical solution to the Riemann problem, based on the method of characteristics, is developed by introducing a gas selectivity ratio representing the gas relative sorption affinity. The influence of gas selectivity ratio on the enhanced coalbed methane processes is identified.

ON THE NONLINEAR RIEMANN PROBLEMS FOR GENERALFIRST ELLIPTIC SYSTEMS IN THE PLANE

The nonlinear Riemann problem for general systems of the first-order linear and quasi-linear equations in the plane are considered.It translates them to singular integral equations and proves the existence of the solution by means of contract principle or general contract principle.The known results are generalized.