Classification of Codimension-One Riemann Solutions

We investigate solutions of Riemann problems for systems of two conservation laws in one spatial dimension. Our approach is to organize Riemann solutions into strata of successively higher codimension. The codimension-zero stratum consists of Riemann solutions that are structurally stable: the number and types of waves in a solution are preserved under small perturbations of the flux function and initial data. Codimension-one Riemann solutions, which constitute most of the boundary of the codimension-zero stratum, violate structural stability in a minimal way. At the codimension-one stratum, either the qualitative structure of Riemann solutions changes or solutions fail to be parameterized smoothly by the flux function and the initial data. In this paper, we give an overview of the phenomena associated with codimension-one Riemann solutions. We list the different kinds of codimension-one solutions, and we classify them according to their geometric properties, their roles in solving Riemann problems, and their relationships to wave curves.     

Numerical modeling of multiphase first-contact miscible flows. Part 1. Analytical Riemann solver

In this series of two papers, we present a front-tracking method for the numerical simulation of first-contact miscible gas injection processes. The method is developed for constructing very accurate (or even exact) solutions to one-dimensional initial-boundary-value problems in the form of a set of evolving discontinuities. The evolution of the discontinuities is given by analytical solutions to Riemann problems. In this paper, we present the mathematical model of the problem and the complete Riemann solver, that is, the analytical solution to the one-dimensional problem with piecewise constant initial data separated by a single discontinuity, for any left and right states. The Riemann solver presented here is the building block for the front-tracking/streamline method described and applied in the second paper.     

The Riemann problem for a hyperbolic model of two-phase flow in conservative form

This article is to continue the present author's work (International Journal of Computational Fluid Dynamics (2009) 23 (9), 623–641) on studying the structure of solutions of the Riemann problem for a system of three conservation laws governing two-phase flows. While existing solutions are limited and found quite recently for the Baer and Nunziato equations, this article presents the first instance of an exact solution of the Riemann problem for two-phase flow in gas–liquid mixture. To demonstrate the structure of the solution, we use a hyperbolic conservative model with mechanical equilibrium and without velocity equilibrium. The Riemann problem solution for the model equations comprises a set of elementary waves, rarefaction and discontinuous waves of various types. In particular, such a solution treats both the wave structure and the intermediate states of the two-phase gas–liquid mixture. The resulting exact Riemann solver is fully non-linear, direct and complete. On this basis then, we use locally the exact Riemann solver for the two-phase flow in gas–liquid mixture within the framework of finite volume upwind Godunov methods. In order to demonstrate the effectiveness and accuracy of the proposed solver, we consider a series of test problems selected from the open literature and compare the exact and numerical results by using upwind Godunov methods, showing excellent oscillation-free results in two-phase fluid flow problems.     

Direct Riemann solvers for the time-dependent Euler equations

Approaches for finding direct, approximate solutions to the Riemann problem are presented. These result in three approximate Riemann solvers. Here we discuss the time-dependent Euler equations but the ideas are applicable to other systems. The approximate solvers are (i) assessed on local Riemann problems with exact solutions and (ii) used in conjunction with the Weighted Average Flux (WAF) method to solve the two-dimensional Euler equations numerically. The resulting numerical technique is assessed on a shock reflection problem. Comparison with experimental observation is carried out.     

Restoration of the contact surface in the HLL-Riemann solver

The missing contact surface in the approximate Riemann solver of Harten, Lax, and van Leer is restored. This is achieved following the same principles as in the original solver. We also present new ways of obtaining wave-speed estimates. The resulting solver is as accurate and robust as the exact Riemann solver, but it is simpler and computationally more efficient than the latter, particulaly for non-ideal gases. The improved Riemann solver is implemented in the second-order WAF method and tested for one-dimensional problems with exact solutions and for a two-dimensional problem with experimental results.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

WAVE INTERACTIONS IN SULICIU MODEL FOR DYNAMIC PHASE TRANSITIONS

Elementary waves in Suliciu model for dynamic phase transitions are obtained through traveling wave analysis.For any given initial data with two pieces of constant states,the Riemann solutions are constructed as a combination of elementary waves. When the initial profile contains three pieces of constant states,the solution may be constructed from the Riemann solutions,with each two adjacent states connected by elementary waves.A new Riemann problem forms when these two waves collide.Through the exploration of these Riemann problems,the outcome of wave interactions may be classified in a suitable parametric space.     

Riemann–Hilbert method and multi-soliton solutions of the Kundu-nonlinear Schrödinger equation

In this work, we study the Kundu-nonlinear Schrödinger (Kundu-NLS) equation (so-called the extended NLS equation), which can describe the propagation of the waves in dispersive media. A Lax spectral problem is used to construct the Riemann–Hilbert problem, via a matrix transformation. Based on the inverse scattering transformation, the general solutions of the Kundu-NLS equation are calculated. In the reflection-less case, the special matrix Riemann–Hilbert problem is carefully proposed to derive the multi-soliton solutions. Finally, some novel dynamics behaviors of the nonlinear system are theoretically and graphically discussed.

     

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-Dring (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...     

Four-component gas/water/oil displacements in one dimension: part II,example solutions

In this article analytical solutions are constructed for a system of conservation laws modeling compositional flow of four components in three phases using the method of characteristics (MOC). Every component partitions between all phases present, and the equilibrium volume ratios of the components in each phase are fixed. Riemann problems modeling the injection of carbon dioxide and water in a depleted oil reservoir are studied, and the sensitivity of the solutions to changes in boundary conditions is analyzed. Finally the MOC solutions are compared to simulated displacements.     

Quasi‐One‐Dimensional Riemann Problems and Their Role in Self‐Similar Two‐Dimensional Problems

We study two‐dimensional Riemann problems with piecewise constant data. We identify a class of two‐dimensional systems, including many standard equations of compressible flow, which are simplified by a transformation to similarity variables. For equations in this class, a two‐dimensional Riemann problem with sectorially constant data becomes a boundary‐value problem in the finite plane. For data leading to shock interactions, this problem separates into two parts: a quasi‐one‐dimensional problem in supersonic regions, and an equation of mixed type in subsonic regions. We prove a theorem on local existence of solutions of quasi‐one‐dimensional Riemann problems. For 2 × 2 systems, we generalize a theorem of Courant & Friedrichs, that any hyperbolic state adjacent to a constant state must be a simple wave. In the subsonic regions, where the governing equation is of mixed hyperbolic‐elliptic type, we show that the elliptic part is degenerate at the boundary, with a nonlinear variant of a degeneracy first described by Keldysh. (Accepted December 4, 1997)     

Calculation of viscous and inviscid gas flows on unstructured grids using the AUSM scheme

An approach to the solution of the two-dimensional Navier-Stokes equations on triangular unstructured grids is considered. The method is based on the key idea of the Godunov scheme, namely, the advisability of solving the Riemann problem of arbitrary discontinuity breakdown. In the calculations the derivatives with respect to space are approximated with both the first and the second order. However, as distinct from the conventional Godunov method, in calculating the fluxes across the cell boundaries the Riemann problem is solved using the Advection Upstream Splitting Method (AUSM). The concepts involved in the AUSM scheme are discussed. The solution of the discontinuity breakdown problem obtained within the framework of this approach is compared with the results obtained using the Godunov method. Numerical solutions of some problems of viscous and inviscid perfect-gas flows obtained on unstructured grids of different fineness and those obtained on structured grids are also compared. The effect of the spatial approximation order on the accuracy of numerical solutions is studied.     

Performance of finite volume solutions to the shallow water equations with shock‐capturing schemes

Numerical methods have become well established as tools for solving problems in hydraulic engineering. In recent years the finite volume method (FVM) with shock capturing capabilities has come to the fore because of its suitability for modelling a variety of types of flow; subcritical and supercritical; steady and unsteady; continuous and discontinuous and its ability to handle complex topography easily. This paper is an assessment and comparison of the performance of finite volume solutions to the shallow water equations with the Riemann solvers; the Osher, HLL, HLLC, flux difference splitting (Roe) and flux vector splitting. In this paper implementation of the FVM including the Riemann solvers, slope limiters and methods used for achieving second order accuracy are described explicitly step by step. The performance of the numerical methods has been investigated by applying them to a number of examples from the literature, providing both comparison of the schemes with each other and with published results. The assessment of each method is based on five criteria; ease of implementation, accuracy, applicability, numerical stability and simulation time. Finally, results, discussion, conclusions and recommendations for further work are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

On the Energy Dissipation Rate of Solutions to the Compressible Isentropic Euler System

In this paper we extend and complement the results in Chiodaroli et al. (Global ill-posedness of the isentropic system of gas dynamics, 2014) on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law p(ρ) = ρ ^γ , γ ≥ 1. First we show that every Riemann problem whose one-dimensional self-similar solution consists of two shocks admits also infinitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Székelyhidi (Ann Math 170:1417–1436, 2009), (Arch Ration Mech Anal 195:225–260, 2010). Moreover we prove that for some of these Riemann problems and for 1 ≤ γ < 3 such solutions have a greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos in (J Diff Equ 14:202–212, 1973) does not favour the classical self-similar solutions.     

On the Riemann problem for liquid or gas-liquid media

Self-similar solutions to the Riemann problem for water with the modified Tait equation of state are presented. The methods of Smoller for gas dynamics are employed to reduce the problem to the solution of a single non-linear equation. The same methods are used for solving the Riemann problem at a gas-water interface. In both cases the method of interval bisections affords a solution technique free of problems with convergence.     

Codimension-One Riemann Solutions: Missing Rarefactions Adjacent to Doubly Sonic Transitional Waves

This paper is the fifth in a series that undertakes a systematic investigation of Riemann solutions of systems of two conservation laws in one spatial dimension. In this paper, three degeneracies that can occur only in Riemann solutions that contain doubly sonic transitional shock waves, together with the degeneracies that pair with them, are studied in detail. Conditions for a codimension-one degeneracy are identified in each case, as are conditions for folding of the Riemann solution surface. Simple examples are given, including a numerically computed Riemann solution that contains a doubly sonic transitional shock wave.     

HLLC scheme for the preconditioned pseudo-compressibility Navier–Stokes equations for incompressible viscous flows

A new HLLC (Harten-Lax-van leer contact) approximate Riemann solver with the preconditioning technique based on the pseudo-compressibility formulation for numerical simulation of the incompressible viscous flows has been proposed, which follows the HLLC Riemann solver (Harten, Lax and van Leer solver with contact resolution modified by Toro) for the compressible flow system. In the authors' previous work, the preconditioned Roe's Riemann solver is applied to the finite difference discretisation of the inviscid flux for incompressible flows. Although the Roe's Riemann solver is found to be an accurate and robust scheme in various numerical computations, the HLLC Riemann solver is more suitable for the pseudo-compressible Navier--Stokes equations, in which the inviscid flux vector is a non-homogeneous function of degree one of the flow field vector, and however the Roe's solver is restricted to the homogeneous systems. Numerical investigations have been performed in order to demonstrate the efficiency and accuracy of the present procedure in both two- and three-dimensional cases. The present results are found to be in good agreement with the exact solutions, existing numerical results and experimental data.     

An exact Riemann solver for detonation products

Numerical methods based upon the Riemann Problem are considered for solving the general initial-value problem for the Euler equations applied to real gases. Most of such methods use an approximate solution of the Riemann problem when real gases are involved. These approximate Riemann solvers do not yield always a good resolution of the flow field, especially for contact surfaces and expansion waves. Moreover, approximate Riemann solvers cannot produce exact solutions for the boundary points. In order to overcome these shortcomings, an exact solution of the Riemann problem is developed, valid for real gases. The method is applied to detonation products obeying a 5th order virial equation of state, in the shock-tube test case. Comparisons between our solver, as implemented in Random Choice Method, and finite difference methods, which do not employ a Riemann solver, are given.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Riemann solvers with evolved initial conditions

The scope of this paper is three fold. We first formulate upwind and symmetric schemes for hyperbolic equations with non‐conservative terms. Then we propose upwind numerical schemes for conservative and non‐conservative systems, based on a Riemann solver, the initial conditions of which are evolved non‐linearly in time, prior to a simple linearization that leads to closed‐form solutions. The Riemann solver is easily applied to complicated hyperbolic systems. Finally, as an example, we formulate conservative schemes for the three‐dimensional Euler equations for general compressible materials and give numerical results for a variety of test problems for ideal gases in one and two space dimensions. Copyright © 2006 John Wiley & Sons, Ltd.     

Integral equations of plane static boundary value problems of the elasticity theory for an inhomogeneous anisotropic medium

The static boundary value problems of plane elasticity for an inhomogeneous anisotropic medium in a simply connected domain are reduced to the Riemann–Hilbert problem for a quasianalytic vector. Singular integral equations over the domain are obtained, and their solvability is proved for a sufficiently wide anisotropy class. In the case of a homogeneous anisotropic body, the solutions of the first and second boundary value problems are obtained in closed form.     

Lattice Boltzmann methods for shallow water flow applications

We apply the lattice Boltzmann (LB) method for solving the shallow water equations with source terms such as the bed slope and bed friction. Our aim is to use a simple and accurate representation of the source terms in order to simulate practical shallow water flows without relying on upwind discretization or Riemann problem solvers. We validate the algorithm on problems where analytical solutions are available. The numerical results are in good agreement with analytical solutions. Furthermore, we test the method on a practical problem by simulating mean flow in the Strait of Gibraltar. The main focus is to examine the performance of the LB method for complex geometries with irregular bathymetry. The results demonstrate its ability to capture the main flow features. Copyright © 2007 John Wiley & Sons, Ltd.