Computation of Riemann solutions using the Dafermos regularization and continuation

We investigate the solution of the nonlinear junction Riemann problem for the one-dimensional shallow water equations (SWEs) in a simple star network made of three rectangular channels. We consider possible bottom discontinuities between the channels and possible differences in the channels width. In the literature, the solution of the Riemann problem at the junction is investigated for the symmetric case without bottom steps and channels width variations. Here, the solution is extended to a more general situation such that neither the equality of the channels width nor the symmetry of the flow are assumed in the downstream channels. The analysis is performed under sub-criticality conditions and the results are summarized in a main theorem, while a series of numerical examples are presented and support our conclusions.

The numerical approximation of liquid vapor flows within the compressible regime is a challenging task because complex physical effects at the phase interfaces govern the global flow behavior. We develop a sharp interface approach which treats the phase boundary like a shock wave discontinuity and takes capillarity effects into account. The approach relies on the solution of Riemann problems across the interface that separates the liquid and the vapor phase. The Riemann solution accounts for the relevant physics by enforcing appropriate jump conditions at the phase boundary. A wide variety of interface effects can be handled in a thermodynamically consistent way. This includes surface tension, as well as, mass and energy transfer by phase transition. Moreover, the local normal speed of the interface, which is needed to calculate the time evolution of the phase boundary, is given by the Riemann solution. The focus of this work is the development of isothermal and non-isothermal two-phase Riemann solvers for the sharp interface approach. To verify the solvers with respect to numerical and thermodynamic requirements, one-dimensional and radially symmetric problems are studied. Furthermore, the Riemann solvers and the sharp interface approach are successfully validated against shock tube experiments of real fluids (alkanes).

Solution of Riemann problem for dusty gas flow

We consider a system of two nonhyperbolic conservation laws modelling incompressible two-phase flow in one space dimension. The purpose of this paper is to justify the use of singular shocks in the solution of Riemann problems. We prove that both strictly and weakly overcompressive singular shocks are limits of viscous structures. Using Riemann solutions we solve Cauchy problems with piecewise constant data for the nonhyperbolic two-fluid model.

We consider a nonlinear hyperbolic model describing phase transitions in nonlinear elastodynamics. The Riemann problem is solved uniquely, provided we supplement the fundamental conservation laws (mass, momentum) with a kinetic relation. The latter takes into account small-scale mechanisms, such as the viscosity and capillary effects in the material under consideration. Our construction generalizes, to solutions of arbitrary large amplitude of the model under study, an approach proposed by Hayes and LeFloch for general systems of conservation laws. The Riemann solutions may contain rarefaction waves, (compressive) classical shock waves, as well as (undercompressive) nonclassical shocks.

The Riemann problem for two-dimensional gas dynamics with isentropic or polytropic gas is considered. The initial data is constant in each quadrant and chosen so that only a rarefaction wave, shock wave, or slip line connects two neighboring constant initial states. With this restriction sixteen (respectively, fifteen) genuinely different wave combinations for isentropic (respectively, polytropic) gas exist. For each configuration the numerical solution is analyzed and illustrated by contour plots. Additionally, the required relations for the initial data and the symmetry properties of the solutions are given. The chosen calculations correspond closely to the cases studied by T. Zhang and Y. Zheng [SIAM J. Math. Anal., 21 (1990), pp. 593–630], so that the analytical theory can be directly compared to our numerical study.

Traveling Waves, Riemann Problems and Computations of a Model of the Dynamics of Liquid/Vapor Phase Transitions

In this paper, we are concerned with the Riemann problem and the perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction, which can also be seen as the nonsymmetric Keyfitz–Kranzer system with Chaplygin pressure and Coulomb-like friction. For the Riemann problem, we show that it explicitly exhibits two kinds of different structures and the delta shock wave appears in some certain situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the Coulomb-like friction term makes contact discontinuities and the delta shock wave bend into parabolic shapes and the Riemann solutions are not self-similar anymore. For the perturbed Riemann problem with delta initial data, not only the delta shock wave but also the delta contact discontinuity are found in solutions and the friction term makes them bent. Under the generalized Rankine–Hugoniot conditions and the entropy condition, by taking variable substitution, we constructively obtain the global existence of generalized solutions which explicitly exhibit four kinds of different structures. The results in this paper yield a way of studying the wave interaction involving the delta shock wave for conservation laws with source terms and will give us some insights into the research on the Chaplygin pressure Aw–Rascle model, the pressureless Euler equations and the Chaplygin Euler equations with various kinds of source terms.

A numerical technique, called a ‘weighted average flux’ (WAF) method, for the solution of initial-value problems for hyperbolic conservation laws is presented. The intercell fluxes are defined by a weighted average through the complete structure of the solution of the relevant Riemann problem. The aim in this definition is the achievement of higher accuracy without the need for solving ‘generalized’ Riemann problems or adding an anti-diffusive term to a given first-order upwind method. Second-order accuracy is proved for a model equation in one space dimension; for nonlinear systems second-order accuracy is supported by numerical evidence. An oscillation-free formulation of the method is easily constructed for a model equation. Applications of the modified technique to scalar equations and nonlinear systems gives results of a quality comparable with those obtained by existing good high resolution methods. An advantage of the present method is its simplicity. It also has the potential for efficiency, because it is well suited to the use of approximations in the solution of the associated Riemann problem. Application of WAF to multidimensional problems is illustrated by the treatment using dimensional splitting of a simple model problem in two dimensions.

ABSTRACTThe exact solutions for the Riemann problem concerning the one‐dimensional triple‐pressure Euler equations with the Coulomb‐type frictional term are displayed in perfectly explicit forms, where both the rarefaction and shock waves are presented in parabolic shapes with equal curvature under the action of the Coulomb‐type frictional term. Specifically, the curved delta shock wave is formed by sending the limit of Riemann solution comprised of double shock waves, and the vacuum state is also grown up by taking the limit of Riemann solution comprised of double rarefaction waves when all the three perturbation parameters are dropped to zero, where the remarkable concentration and cavitation phenomena can be closely observed and explored. Besides, the numerical simulations in correspondence are also offered to validate our results.

The Riemann problem for the Umami Chaplygin Aw-Rascle model of traffic flow with Coulomb-like friction is solved by introducing the reasonable flux-function. Three kinds of non-self-similar Riemann solutions involving delta-shock are given exactly by the means of variable substitution method. The serious traffic congestion may be explained by the delta-shock, for which, the generalized Rankine–Hugoniot relation and entropy condition are clarified. Since the friction term is involved, all waves containing the rarefaction wave, contact discontinuity, shock wave, and delta-shock are bent into parabolic curves. By analyzing the limiting behavior of the solutions, it is discovered that as the Umami Chaplygin gas pressure and flux-function vanish simultaneously, the Riemann solution containing the rarefaction wave and the contact discontinuity tends to the vacuum solution of the zero-pressure Euler equations with friction, and the solution including the shock wave and the contact discontinuity tends to the delta-shock solution of the zero-pressure Euler equations with friction.

Derivative Riemann solvers for systems of conservation laws and ADER methods

We give a solution of the Riemann problem in relativistic hydrodynamics in the case of ultrarelativistic equation of state and nonvanishing components of the velocity tangent to the initial discontinuity. Simplicity of the ultrarelativistic equation of state (the pressure being directly proportional to the energy density) allows us to express this solution in analytical terms. The result can be used both to construct and test numerical schemes for relativistic Euler equations in (3+1) dimensions.

The Riemann problem and a Godunov-type scheme for a traffic flow model on two lanes with two velocities

We analyze the appearance of delta shock wave and vacuum state in the vanishing pressure limit of Riemann solutions to the non-isentropic generalized Chaplygin gas equations. As the pressure vanishes, the Riemann solution including two shock waves and possible one contact discontinuity converges to a delta shock wave solution. Both the densityρand the internal energyHsimultaneously present a Dirac delta singularity. And the Riemann solution involving two rarefaction waves and possible one contact discontinuity converges to a solution involving vacuum state of the transport equations.