Riemann Problem For Hyperbolic System Research Articles

On a Two-Dimensional Riemann Problem for Hyperbolic System of Nonlinear Conservation Laws

On a Two-Dimensional Riemann Problem for Hyperbolic System of Nonlinear Conservation Laws

Shocks make the Riemann problem for the full Euler system in multiple space dimensions ill-posed

The question of (non-)uniqueness of one-dimensional self-similar solutions to the Riemann problem for hyperbolic systems of gas dynamics in the class of multi-dimensional admissible weak solutions was addressed in recent years in several papers culminating in [] with the proof that the Riemann problem for the isentropic Euler system with a power law pressure is ill-posed if the one-dimensional self-similar solution contains a shock. Then the natural question arises whether the same holds also for a more involved system of equations, the full Euler system. After the first step in this direction was made in [], where ill-posedness was proved in the case of two shocks appearing in the self-similar solution, we prove in this paper that the presence of just one shock in the self-similar solution implies the same outcome, i.e. the existence of infinitely many admissible weak solutions to the multi-dimensional problem.

EXACT RIEMANN SOLUTIONS TO COMPRESSIBLE EULER EQUATIONS IN DUCTS WITH DISCONTINUOUS CROSS-SECTION

We determine completely the exact Riemann solutions for the system of Euler equations in a duct with discontinuous varying cross-section. The crucial point in solving the Riemann problem for hyperbolic system is the construction of the wave curves. To address the difficulty in the construction due to the nonstrict hyperbolicity of the underlying system, we introduce the L-M and R-M curves in the velocity-pressure phase plane. The behaviors of the L-M and R-M curves for six basic cases are fully analyzed. Furthermore, we observe that in certain cases the L-M and R-M curves contain the bifurcation which leads to the nonuniqueness of the Riemann solutions. Nevertheless, all possible Riemann solutions including classical as well as resonant solutions are solved in a uniform framework for any given initial data.

Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms

We compare four different approximate solvers for the generalized Riemann problem (GRP) for non-linear systems of hyperbolic equations with source terms. The GRP is a special Cauchy problem for a hyperbolic system with source terms whose initial condition is piecewise smooth. We briefly review the four solvers currently available and carry out a systematic assessment of these in terms of accuracy and computational efficiency. These solvers are the building block for constructing high-order numerical schemes of the ADER type for solving the general initial-boundary value problem for inhomogeneous systems in multiple space dimensions, in the frameworks of finite volume and discontinuous Galerkin finite element methods.

An iterative Riemann solver for systems of hyperbolic conservation laws, with application to hyperelastic solid mechanics

In this paper, we present a general iterative method for the solution of the Riemann problem for hyperbolic systems of PDEs. The method is based on the multiple shooting method for free boundary value problems. We demonstrate the method by solving one-dimensional Riemann problems for hyperelastic solid mechanics. Even for conditions representative of routine laboratory conditions and military ballistics, dramatic differences are seen between the exact and approximate Riemann solution. The greatest discrepancy arises from misallocation of energy between compressional and thermal modes by the approximate solver, resulting in nonphysical entropy and temperature estimates. Several pathological conditions arise in common practice and modifications to the method to handle these are discussed. These include points where genuine nonlinearity is lost, degeneracies, and eigenvector deficiencies that occur upon melting.

The general multi-dimensional Riemann problem for hyperbolic systems with real constant coefficients

In this paper, we give the explicit solution to the general multi-dimensional Riemann problem for the canonical form of $2\times 2$ hyperbolic systems with real constant coefficients.

LINEARIZED FORMULATION OF THE RIEMANN PROBLEM FOR RADIATION HYDRODYNAMICS

The solution of the Riemann problem is one of the fundamental ingredients in theprocess of building higher-order Godunov schemes for the numerical solution of hyperbolic problems. In Balsara ( JQSRT, 1999, 61, 617–627) we were able to show that the equations ofradiation hydrodynamics can be viewed as a hyperbolic system of equations. As suggested byRoe ( J. Comput. Phys., 1981, 43, 357), much of the information generated in the process ofsolving the Riemann problem for hyperbolic systems is actually never used in the constructionof upwinded fluxes. This fact prompts the development of linearized formulations of theRiemann problem. In this paper we formulate such a linearized Riemann solver for theequations of radiation hydrodynamics. It is shown that a consistent state exists. The consistentstate that we have found also satisfies Roe’s Property U. As a result, isolated strong shocks ofarbitrary strength are exactly modeled by the Riemann solver. It is also shown that thisconsistent state can be used to advantage in evaluating the eigenvalues and eigenvectors thatresult from the linearization of the Riemann problem for radiation hydrodynamics.

Riemann problem for hyperbolic systems of conservation laws with no classical wave solutions

Riemann problem for hyperbolic systems of conservation laws with no classical wave solutions

The Solution of the Riemann Problem for a Hyperbolic System of Conservation Laws Modeling Polymer Flooding

The global Riemann problem for a nonstrictly hyperbolic system of conservation laws modeling polymer flooding is solved. In particular, the system contains a term that models adsorption effects.

Existence and uniqueness of entropy solutions to the Riemann problem for hyperbolic systems of two nonlinear conservation laws

This paper contains a proof of the existence and uniqueness of solutions to the Riemann problem for systems of two hyperbolic conservation laws in one space variable. Our main assumptions are that the system is strictly hyperbolic and genuinely nonlinear. We also require that the system satisfy standard conditions on the second Fréchet derivatives, and one other hypothesis, which we have called the half-plane condition. This hypothesis replaces other, more restrictive hypotheses required by previous authors. The methods and results of this paper are designed to be applicable to systems of conservation laws which are not strictly hyperbolic.

Structure of solutions of the Riemann problem for hyperbolic systems of conservation laws

Structure of solutions of the Riemann problem for hyperbolic systems of conservation laws

Solutions of the Riemann problem for hyperbolic systems of quasilinear equations without convexity conditions

Solutions of the Riemann problem for hyperbolic systems of quasilinear equations without convexity conditions