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This work focuses on the analytical solution to the Riemann problem (RP) for a 1-D non-ideal flow of dusty gas with external force. Here it is presumed that external force is a continuous function of time. We explicitly obtain the elementary wave curves to 1-D non-ideal flow of dusty gas with external force and determine these wave curves in form of characteristics. Exhaustive calculations were performed for the elementary wave solutions, such as the rarefaction wave, shock wave, and contact discontinuity. We examine the influence of dust particles on density, velocity of flow, and shock speed and their implications on the solution of RP. Also, the implication of addition of external force is that all solutions are not self similar. [ABSTRACT FROM AUTHOR]

This work proposes a mathematical model to study the foam displacement in porous media stabilized by nanoparticles. We consider a simplification of the Stochastic Bubble Population balance model in local equilibrium, with nanoparticle dependence inspired by the experimental data from the literature. It consists of a non-strictly hyperbolic system of conservation laws, which is solved for the generic initial and injection conditions. We investigate the existence of a global solution as a sequence of waves following the Conservation Laws Theory. When the solution is composed of two or more waves, we present necessary and sufficient conditions to guarantee the compatibility of these wave sequences. The analytical solution for the nanoparticle-stabilized foam displacement in porous media allowed us to quantify the effect of nanoparticles on foam displacement, focusing on the breakthrough time and cumulative water production. In agreement with the literature, when only gas is injected, the breakthrough time and the water production increase with the nanoparticle concentration. Although, we also observe that the effect of nanoparticles is less pronounced for high nanoparticle concentration. Counterintuitively, during gas-water co-injection for a certain parameter range, adding nanoparticles changes the mathematical solution qualitatively, yielding a negligible effect on water production. We discuss the most favorable conditions to observe the action of nanoparticles in laboratory experiments. • Recent studies reported increasing foam stability when employing nanoparticles. • We model the nanoparticle-stabilized foam flow allowing an analytical solution. • We solved the corresponding non-strictly hyperbolic system of conservation laws. • Counterintuitively, adding nanoparticles can lead to the same water production. [ABSTRACT FROM AUTHOR]

Construction of Riemann solutions for a hyperbolic system in conservative form arising from the one-dimensional mean-field games with the quadratic Hamiltonian and the logarithmic coupling term are provided in detail. Moreover, the delta shock formation is concretely analyzed from the limit of double-shock-solution as well as the emergence of vacuum state is also specifically discussed from the limit of double-rarefaction-solution when the coupling coefficient drops to zero. Accordingly, the remarkable cavitation and concentration phenomena can be closely observed and explored. Additionally, the numerical experiments are also presented in correspondence to authenticate the theoretical analysis results. • Riemann solutions for a hyperbolic system from the mean-field games are constructed. • Delta shock formation is analyzed from the limit of double-shock-solution. • The emergence of vacuum state is discussed from the limit of double-rarefaction-solution. • Cavitation and concentration phenomena are observed and explored. [ABSTRACT FROM AUTHOR]

Two kinds of exact Riemann solutions for a hyperbolic system arising from the steady 2D Helmholtz equation under a paraxial assumption are constructively achieved by using either delta shock wave or contact-vacuum-contact composite wave, which depends on the ordering relation between the left and right initial velocities. Moreover, the interaction between delta shock wave and contact-vacuum-contact composite wave is carefully explored by considering the initial data in three pieces separated by two jump discontinuities. [ABSTRACT FROM AUTHOR]

In this paper, we study the Riemann problem for a general variable coefficient Burgers equation with time-dependent damping. We use a nonlinear time-dependent viscosity equation with a similarity variable. Thus, when the viscosity goes to zero, we obtain Riemann solutions to the general variable coefficient Burgers equation with time-dependent damping. Moreover, we use the Lax–Friedrichs scheme to obtain numerical evidence of the Riemann solutions. • A general variable coefficient Burgers equation (vc-B) with damping is studied. • Riemann solutions to the vc-B with time-dependent damping are constructed. • Mathematical comprehension of Riemann solutions to vc-B with time-dependent damping. • We propose a generalized Dafermos regularization and Riemann solutions are found. • We present simulations to obtain numerical evidence of the theoretical solutions. [ABSTRACT FROM AUTHOR]

In this paper, the Riemann solutions for the macroscopic production model with chaplygin gas under some special initial data are constructively obtained in the fully explicit form. An interesting composite wave R δ J is observed, it is formed by a rarefaction wave R and a left-contact delta discontinuity δ J attached to the wavefront of the rarefaction wave. Furthermore, this delta discontinuity gradually absorbs the rarefaction wave and eventually forms a delta shock wave under some suitable initial condition. In addition, we give some typical numerical results that coincide well with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper we propose a modeling setting and a numerical Riemann problem solver at the junction of one dimensional shallow-water channel networks. The junction conditions take into account the angles with which the channels intersect and include the possibility of channels with different sections. The solver is illustrated with several numerical tests which underline the importance of the angle dependence to obtain reliable solutions. [ABSTRACT FROM AUTHOR]

We present a novel methodology based on Physics-Informed Neural Networks (PINNs) for solving systems of partial differential equations admitting discontinuous solutions. Our method, called Gradient-Annihilated PINNs (GA-PINNs), introduces a modified loss function that forces the model to partially ignore high-gradients in the physical variables, achieved by introducing a suitable weighting function. The method relies on a set of hyperparameters that control how gradients are treated in the physical loss. The performance of our methodology is demonstrated by solving Riemann problems in special relativistic hydrodynamics, extending earlier studies with PINNs in the context of the classical Euler equations. The solutions obtained with the GA-PINN model correctly describe the propagation speeds of discontinuities and sharply capture the associated jumps. We use the relative l 2 error to compare our results with the exact solution of special relativistic Riemann problems, used as the reference "ground truth", and with the corresponding error obtained with a second-order, central, shock-capturing scheme. In all problems investigated, the accuracy reached by the GA-PINN model is comparable to that obtained with a shock-capturing scheme, achieving a performance superior to that of the baseline PINN algorithm in general. An additional benefit worth stressing is that our PINN-based approach sidesteps the costly recovery of the primitive variables from the state vector of conserved variables, a well-known drawback of grid-based solutions of the relativistic hydrodynamics equations. Due to its inherent generality and its ability to handle steep gradients, the GA-PINN methodology discussed in this paper could be a valuable tool to model relativistic flows in astrophysics and particle physics, characterized by the prevalence of discontinuous solutions. [ABSTRACT FROM AUTHOR]

In our present work, we analyze the limiting behaviour of solution to the Riemann problem for a macroscopic production model with van der Waals equation of state. We construct the solution to the Riemann problem of the governing system which consists of only classical elementary waves and observe vacuum state for certain initial data. In the limiting process we establish the formation of extreme concentration for a state variable in terms of Dirac delta distribution. Further it is observed that the delta shock solution of the governing system is different from that of pressureless gas dynamics system so a perturbation to the flux is made and the intrinsic phenomena of concentration and cavitation is examined in the limiting case. Additionally we perform numerical simulations to note the effect of van der Waals parameter on the solution of the Riemann problem and to observe the formation of delta shock and vacuum state in the limiting cases. • A macroscopic production model is investigated. • Limiting behaviour of the solution of the Riemann problem is successfully developed and validated. • The influence of delta shock wave is explored. • Successful verification of the results of the proposed solution in a special case. [ABSTRACT FROM AUTHOR]

In the mathematical modeling of solid fuel combustion, thermal losses are commonly neglected to allow a rigorous analysis. In this work, different choice of simplifications was made, making the model more physically realistic. Besides the analysis of the solutions including existence and uniqueness, a classification in the combustion regimes as a function of thermal losses was obtained. Some examples were analyzed, evidencing the validity of the proposed classification. Numerical simulations with more realistic models show the validity of the proposed formulas and applications for filtration combustion. [ABSTRACT FROM AUTHOR]

In this paper, we study the exact solutions of Riemann problem for the compressible ideal fluid flow, where the external force is a continuous function of time. The exact expressions for contact discontinuity curve, shock wave curve and rarefaction wave curve are provided. Six types of exact solutions of the Riemann problem are also obtained. In particular, a vacuum arises at t > 0 although the initial data never involves the vacuum, and these solutions do not possess self-similarity in the (t , x) -plane. [ABSTRACT FROM AUTHOR]

This paper proposes an invariant-domain preserving approximation technique for nonlinear conservation systems that is high-order accurate in space and time. The algorithm mixes a high-order finite element method with an invariant-domain preserving low-order method that uses the closest neighbor stencil. The construction of the flux of the low-order method is based on an idea from Abgrall et al. (2017). The mass flux of the low-order and the high-order methods are identical on each finite element cell. This allows for mass preserving and invariant-domain preserving limiting. [ABSTRACT FROM AUTHOR]

The phenomena of concentration and cavitation and the formation of delta shock waves and vacuum states in vanishing pressure limits of solutions to the generalized Chaplygin Euler equations of compressible fluid flow are analyzed. It is proved that, as the pressure vanishes, any two-shock-wave Riemann solution of the generalized Chaplygin Euler equations of compressible fluid flow tends to a delta-shock solution to the transport equations, and the intermediate density between them tends to a weighted δ -measure that forms a delta shock wave; any two-rarefaction-wave solution is shown to tend to two contact discontinuities connecting the constant states and vacuum states, which form a vacuum solution of the transport equations. Moreover, some numerical simulations completely coinciding with the theoretical analysis are presented. [ABSTRACT FROM AUTHOR]

We solve the Riemann problem for the Suliciu relaxation system with the Coulomb-like friction terms containing two constant friction coefficients. The exact Riemann solutions involving delta shock waves and contact discontinuities are obtained constructively. Via deriving the generalized Rankine–Hugoniot relations of delta shock waves in detail, all of the position, propagation speed and strength of the delta shock wave are given concretely under a suitable entropy condition. Based on this, we clearly analyze the influence of the Coulomb-like friction terms on the Riemann solutions of the Suliciu relaxation system. • The Riemann problem for the Suliciu relaxation system with the Coulomb-like friction terms is solved. • The Coulomb-like friction terms introduced in this paper contain two independent friction coefficients. • The exact Riemann solutions involving delta shock waves and contact discontinuities are obtained constructively. • We clearly analyze the influence of the Coulomb-like friction terms on the Riemann solutions of the Suliciu relaxation system. [ABSTRACT FROM AUTHOR]

Observations suggest that the First-In-First-Out (FIFO) principle is usually violated on multi-lane roads. Jin (2017) formulated and solved unifiable multi-commodity kinematic wave models, when different commodities have the same contributions to overall traffic congestion (unifiable) but may travel at different speeds (non-FIFO). However, the construction of unifiable multi-commodity fundamental diagrams and the assumption of concave or convex commodity flow proportion functions are purely mathematical and lack behavioral explanations. Thus the existing formulation is only of pure mathematical and theoretical interests and too complicated for real-world calibration, validation, or applications. In this study, we present a new formulation of unifiable multi-commodity kinematic wave models to address the aforementioned limitations. We first introduce a new variable for the relative speed ratios of different commodities and particularly discuss constant relative speed ratios. The relative speed ratios are physically, behaviorally, and economically meaningful since they characterize drivers' relative aggressiveness, values of times, and other features. We then present unifiable multi-commodity fundamental diagrams based on the relative speed ratios, which can be used to derive the mathematical generating functions in Jin (2017). Then we show that non-FIFO multi-commodity kinematic wave model is a system of non-strictly hyperbolic conservation laws and solve the Riemann problem for a two-commodity system with constant relative speed ratios, in which the commodity flow proportion function is either concave or convex. We also present an empirical evidence for the existence of unifiable multi-commodity fundamental diagrams with constant relative speed ratios, which help to demonstrate the advantage of the new formulation. In summary, the new formulation is as general as the original formulation, but physically more meaningful, theoretically easier to solve, and empirically simpler to calibrate. Finally we conclude the study with discussions on potential future applications. [ABSTRACT FROM AUTHOR]

In the frame of α − solutions defined in the setting of distributional products, the discontinuous solutions to the Riemann problem for a nonlinear chromatography system are constructed. All the discontinuous solutions are obtained within a convenient space of distributions including discontinuous functions and Dirac delta measures. The constructed α − solutions are reasonable in comparison with the known results by using other techniques. [ABSTRACT FROM AUTHOR]

• The Riemann problem at channel obstructions and constrictions is solved. • The mathematical solution makes use of curves that generalize the 1- and 2-waves. • The solution exists and it is unique for a wide class of initial flow conditions. • The exact solutions shed light on the numerical issues of current 1D models. • A novel 1D numerical model is based on the local solution of a Riemann problem. Usually, the rapid geometric transitions that are of negligible length with respect to the channel are treated in one-dimensional Saint Venant models as internal boundary conditions, assuming that an instantaneous equilibrium is attained between the flow characteristics through the structure and the flow characteristics in the channel. In the present paper, a different point of view is assumed by considering rapid transients at channel constrictions and obstructions that are caused by the lack of instantaneous equilibrium between the flow conditions immediately upstream and downstream of the structure. These transients are modelled as a Riemann problem, assuming that the flow through the geometric transition is described by a stationary weak solution of the Saint Venant equations without friction. For this case, it is demonstrated that the solution of the Riemann problem exists and it is unique for a wide class of initial flow conditions, including supercritical flows. The solutions of the Riemann problem supplied by the one-dimensional mathematical model compare well with the results of a two-dimensional Shallow Water Equations numerical model when the head loss through the structure is negligible. The inspection of the exact solutions structure shows that the flow conditions immediately to the left and to the right of the geometric discontinuity may be very different from the initial conditions, and this contributes to explain the numerical issues that are reported in the literature for the rapid transients at internal boundary conditions in finite difference models. The solution of the Riemann problem has been coded, and the corresponding exact fluxes have been used as numerical fluxes in a one-dimensional Finite Volume scheme for the solution of the Shallow water Equations. The results demonstrate that spurious oscillations and instability phenomena are completely eliminated, ensuring the robustness of the approach. In the case that the energy loss is not negligible, the exact solutions capture the essential features of the two-dimensional model numerical results, ensuring that the mathematical procedure is generalizable to realistic conditions. This generalization is presented in the final part of the paper. [ABSTRACT FROM AUTHOR]

The main motive of the present paper is to derive the analytical solution of the Riemann problem for magnetogasdynamic equations governing an inviscid unsteady one-dimensional flow of non-ideal polytropic gas subjected to the transverse magnetic field with infinite electrical conductivity. By using the Lax entropy condition and R–H conditions, we derive the elementary wave solutions i.e. shock wave, simple wave and contact discontinuities without any restriction on the magnitude of initial data states and discussed about their properties. Further, the density and velocity distribution in the flow field for the cases of compressive wave and rarefaction wave is discussed. Here we also compare/contrast the nature of solution in non-ideal magnetogasdynamic flow and ideal gas flow. • Solution of the Riemann Problem for non-ideal magnetogasdynamics flow is obtained. • Lax entropy and R–H conditions are used to derive elementary wave solutions. • Density and velocity profiles for 1- and 3-shock wave is presented. • Effect of non-idealness of the gas in the presence of magnetic field is analyzed. [ABSTRACT FROM AUTHOR]

Abstract The present paper concerns with the study of the Riemann problem for a quasi-linear hyperbolic system of partial differential equations governing the one dimensional isentropic dusty gas flow. The shock and rarefaction waves and their properties for the problem are investigated. We also examine how some of the properties of shock and rarefaction waves in a dusty gas flow differ from isentropic ideal gas flow. The solution of Riemann problem of dusty gas flow for different initial data is discussed. Under certain conditions, the uniqueness and existence of the solution of the Riemann problem has been analyzed. Finally, all possible interactions of elementary waves are discussed. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we concern with the Riemann solutions to a compressible fluid described by the generalized Chaplygin gas, where the external force is a constant. Five exact solutions are given. In particular, the delta shock wave with a Dirac delta function in density and internal energy occurs in some solutions, and the location, velocity and weights of the delta shock wave are explicitly described. It is also noticed that because of the effect of the external force, these exact solutions are not self-similar. Highlights • Five kinds of non-self-similar solutions to a compressible fluid described by the generalized Chaplygin gas are presented. • The delta shock wave with a Dirac delta function in density and internal energy develops in the solutions. • The work provides a fundamental exploration in studying the Navier–Stokes equations with the generalized Chaplygin gas. [ABSTRACT FROM AUTHOR]

• The principal purpose of this paper is the development in numerical treatments for the resonance problem of multi-phase systems by constructing a well-balanced finite volume scheme for two-dimensional shallow water equations with variable topography. • This scheme is constructed based on exact solution of local planar Riemann problems at each grid cells. • The well-balancedness of this scheme is shown in the sense that it can preserve exactly the lake at rest solution over both smooth and non-smooth bottom. • Several numerical experiments related to steady and time-dependent problems are conducted to show the potential of this approach. • The remain challenges are the ineffectiveness of this scheme towards time-dependent solutions throughout a long process and the instability of errors caused by the relaxation while coding. • This work provides an impetus to enhance the performance of this scheme for time-dependent solutions, and motivates further studies in numerical treatments of the resonance problem for multi-phase flow models as well. We consider in this paper a finite volume scheme based on local planar Riemann solutions for the two-dimensional shallow water equations with bathymetry. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. The scheme to be presented is a development of the preliminary works that will be cited below. Our foremost purpose is to extend those results to two-dimensional formalism while preserving the physical and mathematical properties, including the well-balancedness. The proposed scheme is applied to some specific families of solutions, especially lake at rest and partially well-balanced solution. The numerical results show that this approach can give a good accuracy, except for resonant cases. Furthermore, it is proved that our finite volume scheme can preserve the C -property in the sense that it can capture exactly the lake at rest solution. [ABSTRACT FROM AUTHOR]

This study investigates the solutions of the Riemann problem for a two-layered blood flow model which is modeled by a system of quasi-linear hyperbolic partial differential equations (PDEs) obtained by vertically averaging the Euler equations over each layer. We explore the elementary waves, namely shock wave, rarefaction wave and contact discontinuity wave on the basis of method of characteristics. Further, we establish the existence and uniqueness of the corresponding local Riemann solution. Across the contact discontinuity wave, the areas of two nonlinear algebraic equations are determined by using the Newton–Raphson method of two variables in all possible wave combinations. A precise analytical method is used to display a detailed vision of the solution for this model inside a specified space domain and some certain time frame. • The Riemann problem for the two-layered blood flow model in arteries is considered. • The Riemann solution is derived analytically using the method of characteristics. • The properties of elementary waves are analyzed. • The existence and uniqueness of this solution is established. • The physical quantities for all four possible wave combinations are displayed in numerical simulation. [ABSTRACT FROM AUTHOR]

We concern with Riemann problem for an one-dimensional pressureless compressible fluid system with time- and space-dependent external force. Under the concept of α -solution to this model, we obtain the α -solutions to the Riemann problem and the criteria of appearance of each α -solution. It is observed that the expressions of these α -solutions do not depend on α. Besides, the vacuum solution arises although the initial values do not involve vacuum. Furthermore, the delta wave solution emerges for certain initial values, in which internal energy variable and density variable contain the Dirac measure. Finally, our technique can be applied to study the other singular solutions to systems of conservation laws with a source term. • We obtain exactly α -solutions and the criteria of appearance of each α -solution. • The expressions of these α -solutions do not depend on α. • The vacuum solution arises although the initial values do not involve vacuum. • The delta wave solution emerges for certain initial values. [ABSTRACT FROM AUTHOR]

Highlights • Propose a unifiable multi-commodity kinematic wave model. • Construct unifiable commodity speed-density relations. • Analytically solve the Riemann problem for two commodities. • Propose a unifiable multi-commodity Cell Transmission Model. Abstract In the literature, many kinematic wave models have been proposed for multi-class vehicles on multi-lane roads; however, there lacks an explicit model of unifiable multi-commodity traffic, in which different commodity flows can have different speeds and violate the first-in-first-out (FIFO) principle, but there exists a speed-density relation for the total traffic. In this study, we attempt to fill the gap by constructing and solving a unifiable multi-commodity kinematic wave model. We first construct commodity speed-density relations based on generic generating functions. Then for two commodities we discuss the properties of the unifiable kinematic wave model and analytically solve the Riemann problem with a combination of total and commodity kinematic waves. We propose a unifiable multi-commodity Cell Transmission Model (CTM) with a general junction model for numerical simulations of network traffic flows, which are unifiable but may violate the FIFO principle. We prove that the CTM is well-defined under an extended CFL (Courant et al., 1928) condition. With examples we verify the consistency between the analytical and numerical solutions and demonstrate the convergence of the CTM. We conclude with several follow-up research directions for unifiable multi-commodity kinematic wave models. [ABSTRACT FROM AUTHOR]

The Riemann problem for the one-dimensional zero-pressure gas dynamics system is considered in the frame of α − solutions based on a solution concept defined in the setting of a product of distributions. The reformulated form of the zero-pressure gas dynamics system is provided and consequently the unique α − solution is obtained within a convenient class of distributions including the Dirac delta measure. It is shown that our constructed α − solution is reasonable compared with the known results using other methods. Furthermore, the result is generalized for the one-dimensional zero-pressure gas dynamics system with the Coulomb-like friction term, which enables us to see that the α − solution is not self-similar any more. It is shown that the time evolution of the delta shock wave discontinuity is represented by a parabolic curve under the influence of the Coulomb-like friction term. [ABSTRACT FROM AUTHOR]

The Riemann problem for the system of conservation laws of mass, momentum and energy in two-dimensional steady zero-pressure adiabatic flow is solved completely. The Riemann solutions contain two kinds: vacuum states and delta shock waves, on which both density and internal energy simultaneously contain the Dirac delta function. This is quite different from the previous ones on which only one state variable contains the Dirac delta function. The formation mechanism, generalized Rankine–Hugoniot relation and entropy condition are clarified for this type of delta shock wave. Under the suitable generalized Rankine–Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solution is established. In addition, the interactions of delta shock waves and vacuum states are analyzed by solving the Riemann problems with initial data of three piecewise constant states case by case, and the global structure of solutions with four different configurations is constructed. [ABSTRACT FROM AUTHOR]

It has been shown experimentally in the literature that for clayey formations, oil with polar components and an aqueous phase with divalent ions, a secondary waterflood with low salinity water composition improves oil recovery by some 5–20%. Our focus is on a less well known mechanism, i.e. low salinity enhanced solvent (e.g. carbonated water) recovery, as low salinity enhances the aqueous solubility of carbon dioxide. Indeed, after injection the latter is transferred from the aqueous to oleic phase thus decreasing the oil concentration in the oleic phase and diluting the residual oil. To study this mechanism we formulate the conservation equations of total hydrogen, oxygen, chloride and decane. Therefore, we solve analytically and numerically these equations in 1 − D in order to elucidate the effects of the injection of low salinity carbonated water into a reservoir containing oil equilibrated with high salinity carbonated water. We use PHREEQC (acronym of pH-REdox-Equilibrium C-program) to obtain the accurate equilibrium partition of neutral species that are soluble both in the oleic and the aqueous phase by application of the Krichevsky-Ilinskaya extension of Henry's law for solubility of gases in liquids. Using Gibbs phase rule it can be shown that the phase behavior only depends on the pH and the chloride concentration. The above mentioned equilibrium relations use Pitzer's activity coefficients to extend the validity up to 6M. We obtain the saturation, composition and the total Darcy velocity profiles. The significant new insight obtained is that by changing only the salinity in carbonated waterflooding the oil recovery can be enhanced. [ABSTRACT FROM AUTHOR]

The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations with a friction term is considered. Unlike the homogeneous case, the Riemann solutions are no longer self-similar due to the influence of the source term. In particular, delta contact discontinuities are found in the Riemann solutions with delta initial data for the Chaplygin gas equations with the source term. Moreover, we discover that the Chaplygin gas equations can be transformed to the equations of string motion in Minkowski space. The delta shock wave can explain the appearance of the singularities developed in the motion of the string. [ABSTRACT FROM AUTHOR]

We consider the Riemann problem for the system of shallow water equations with horizontal temperature gradients (the Ripa system). The model under investigation has the form of a nonconservative system, and it is hyperbolic, but is not strictly hyperbolic. We construct all solutions of the Riemann problem. It turns out that there may be up to three distinct solutions. A resonant phenomenon which causes the colliding shock waves is observed, where multiple waves associated with different characteristic fields propagate with the same shock speed. [ABSTRACT FROM AUTHOR]

We consider an isothermal gas flowing through a straight pipe and study the effects of a two-way electronic valve on the flow. The valve is either open or closed according to the pressure gradient and is assumed to act without any time or reaction delay. We first give a notion of coupling solution for the corresponding Riemann problem; then, we highlight and investigate several important properties for the solver, such as coherence, consistence, continuity on initial data and invariant domains. In particular, the notion of coherence introduced here is new and related to commuting behaviors of valves. We provide explicit conditions on the initial data in order that each of these properties is satisfied. The modeling we propose can be easily extended to a very wide class of valves. [ABSTRACT FROM AUTHOR]

Zhang and Zheng (1990) conjectured on the structure of a solution for a two-dimensional Riemann problem for Euler equation. To resolve this illuminating conjecture, many researchers have studied the simplified 2 × 2 systems. In this paper, 3-pieces Riemann problem for two-dimensional 2 × 2 hyperbolic system is considered without the restriction that each jump of the initial data projects one planar elementary wave. We classify twelve topologically distinct solutions and construct analytical and numerical solutions. The computed numerical solutions clearly confirm the constructed analytic solutions. [ABSTRACT FROM AUTHOR]

We concern with the Riemann problem the compressible fluid flow with the generalized Chaplygin gas. With the analysis on the phase plane, we rigorously confirm the occurrence of delta shock wave with Dirac delta function in density. Then the formation mechanism, generalized Rankine–Hugoniot relation and entropy condition for the delta shock wave are clarified. Based on these preparations, five kinds of exact solutions are obtained. Finally, the corresponding numerical results are also presented to illustrate our analysis. [ABSTRACT FROM AUTHOR]

The Riemann solutions for a symmetric Keyfitz–Kranzer system are constructed explicitly, in which some singular hyperbolic waves are discovered such as the delta shock wave and the composite wave J R . The global solutions to the double Riemann problem are achieved when the delta shock wave is involved. It is shown that a delta shock wave is separated into a delta contact discontinuity and a shock wave during the process of constructing solutions. [ABSTRACT FROM AUTHOR]

In literature, the von Neumann stability analysis of simplified model equations, such as the wave equation, is typically used to determine stability conditions for the non-linear partial differential fluid flow equations (Navier–Stokes and Euler). However, practical experience suggests that such simplistic stability conditions are grossly inadequate for computations involving the system of coupled flow equations. The goal of this paper is to determine stability conditions for the full system of fluid flow equations – the Euler equations are examined, as any conditions derived for the Euler equations will apply to the Navier–Stokes (NS) equations in the limit of convection-dominated flows. A von Neumann stability analysis is conducted for the one-dimensional (1D) and two-dimensional (2D) Euler equations. The system of equations is discretized on a staggered grid using finite-difference discretization techniques; the use of a staggered grid allows equivalence to finite-volume discretization. By combining the different discretization techniques, ten solution schemes are formulated – eight solution schemes are considered for the 1D Euler equations, and two schemes for the 2D Euler equations. For each scheme, error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum-allowable CFL (Courant–Friedrichs–Lewy) number as a function of Mach number. The predictions are verified for selected schemes using the Riemann problem at incompressible and compressible Mach numbers. Very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, which offer a viable alternative to complicated mathematical expressions often reported in published literature, and should benefit everyday CFD (Computational Fluid Dynamics) users. The stability regions are used to discuss the effect of time integration (explicit vs. implicit), density bias in continuity equation and momentum convection term linearization on stability. A comparison of the predicted stability limits for 1D and 2D Euler equations with commonly-used stability conditions arising from the wave equation shows that the stability thresholds for the Euler equations lie well below those predicted by the wave equation analysis; in addition, the 2D Euler stability limits are more restrictive as compared to 1D Euler limits. Since the present analysis accounts for the full system of fluid flow (Euler) equations, the derived stability conditions can be used by CFD practitioners to estimate a timestep or CFL number to guide the stability of their computations. [ABSTRACT FROM AUTHOR]

This paper presents the development of a cylindrical SPH formulation based on previous study of the literature (Petschek et al) with an explicit formulation for the artificial viscosity. The entire development is explained to propose a formulation adapted to solve Euler equations in the case of a Riemann problem with axis-symmetric conditions. Thus, the artificial viscosity is constructed to find smooth solutions of well-known Riemann problems such as Sod, Noh and Sedov problems. Numerical results are compared to exact solutions and observations are made on numerical parameters influence. This study contributes to validate the axis-symmetrical formulation for pure hydrodynamics tests. [ABSTRACT FROM AUTHOR]

A new simplified analytical model to analyze the penetration of rigid projectiles into soil media is presented. The soil medium is represented by a set of discs, responding in the radial direction under plain strain conditions. A convenient mathematical formulation is derived based on some simplifying assumptions. According to the present approach, the contact parameters in each disc are computed using the developed exact solution of the symmetrical Riemann problem for an irreversible compressible medium. One of the new features of the present model is the incorporation of the exact nonlinear equation of state including unloading-reloading thus considering another key variable that is the maximum medium density that is attained in the process of active loading before unloading is started. Thus the new model considers unloading in the soil medium during the progress of penetration. The present model focuses on the projectile motion and provides information on its velocity, deceleration and depth time histories. It also provides information on the interaction of the projectile with the surrounding soil such as the normal stress distribution along the projectile nose and is capable of determining the contact zone between the nose and the soil. Comparison of the present model results with two-dimensional numerical results as well as with different analytical models and with experimental data is performed. The present model predictions are found to be in good agreement with test data and superior to many existing simplified models. Contrary to many other simplified models, the present model is purely theoretical and does not require any empirical constants or special arbitrary assumptions for calculation of the contact pressures along the projectile nose at all times. The calculations require very small computer time and provide much information regarding the projectile motion in the soil medium. [ABSTRACT FROM AUTHOR]

In this paper, the Riemann problem for the defocusing Hirota equation with weak dispersion is investigated with Whitham modulation theory. Hirota equation can effectively describe the realistic wave motion in dispersive medium. Via averaging Lagrangian method, the Whitham modulation equations in slow modulation form are obtained, which are characterized by wave parameters and reflects the dispersion relation in the original system. Besides, the modulation equations in Riemann invariant form are derived via finite-gap integration theory. Utilizing Whitham modulation equations parameterized by Riemann invariants, the basic structures of solutions for the Riemann problem for original system are acquired. According to the basic structure of the solutions, a complete solution classification corresponding to the initial data is given, including 121 categories. The results are verified by direct numerical simulation. [ABSTRACT FROM AUTHOR]

The exact Riemann solutions are presented in fully explicit forms for the one-dimensional isentropic Euler system of gas dynamics with the body force, in which the shock and rarefaction waves are accelerated into the parabola curves with the same degree under the influence of such body force. Moreover, the limit of Riemann solution composed of two shock waves tends to an accelerated delta shock solution as well as the limit of Riemann solution constituted by two rarefaction waves converges to a solution made up of two contact discontinuities along with the vacuum state encompassed by them when the adiabatic exponent tends to one, in which the intrinsic phenomena of concentration and cavitation can be analyzed and observed carefully. It is of interest to notice that the internal states in two rarefaction wave fans are transformed gradually into the corresponding vacuum states under this limiting circumstance, which is distinguished from the previously established result that a whole rarefaction wave is concentrated into only one contact discontinuity. • Exact Riemann solutions are given for the isentropic Euler system with body force. • Shock and rarefaction waves are accelerated into parabola curves. • Delta shock formation is observed and analyzed as adiabatic exponent goes to one. • The internal states in two rarefaction wave fans become vacuum states gradually. [ABSTRACT FROM AUTHOR]

The Riemann problem and wave interactions are discussed and investigated for an inhomogeneous Aw-Rascle (AR) traffic flow model with extended Chaplygin gas pressure. First, under some variable transformation, the Riemann problem with initial data of two piecewise constants is solved and two different types of Riemann solutions involving rarefaction wave, shock wave and contact discontinuity are obtained. Second, by studying the Riemann problem with three-piecewise-constant initial data, we analyze the interactions of waves and establish the global structures of Riemann solutions. It is shown that, influenced by the source term, the Riemann solutions for the inhomogeneous AR traffic flow model are no longer self-similar, and all the elementary wave curves do not keep straight. Finally, the stability of solution under the small perturbation of initial data is briefly discussed. • The inhomogeneous Aw-Rascle model with extended Chaplygin gas pressure is introduced. • The Riemann problem is solved and the non-self-similar solutions are obtained. • The interactions of waves are investigated. • The stability of solution under the small perturbation of initial data is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we are mainly concerned with the phenomena of cavitation and concentration to the isentropic Euler equations with isothermal dusty gas as the pressure vanishes with double parameters. Firstly, we solve the Riemann problem by analyzing the properties of the elementary waves due to the existence of the inflection points. Secondly, we investigate the limiting behaviors of the Riemann solutions as the pressure vanishes and observe the cavitation and concentration phenomena. Finally, some numerical simulations are performed and the results are consistent with the theoretical analysis. The highlight of this paper is that we extend the restriction of ρ θ ≪ 1 in the previous works to ρ θ < 1 , which makes the wave curve from convex to non-convex. And we prove that the limit of the Riemann solutions of isothermal dusty gas equations is the Riemann solutions of the limit of that equations as pressure vanishes, while the limiting process to vacuum state is different from the previous works. [ABSTRACT FROM AUTHOR]

This work proposes a mathematical model to study the filling up of an unsaturated porous medium by a liquid identifying the transition from unsaturated to saturated flow and allowing a small super saturation. As a consequence the problem remains hyperbolic even when saturation is reached. This important feature enables obtaining numerical solution for any initial value problem and allows employing Glimm’s scheme associated with an operator splitting technique for treating drag and viscous effects. A mixture theory approach is used to build the mechanical model, considering a mixture of three overlapping continuous constituents: a solid (porous medium), a liquid (Newtonian fluid) and a very low-density gas (to account for the mixture compressibility). The constitutive assumption proposed for the pressure gives rise to a continuous function of the fluid fraction. The complete solution of the Riemann problem associated with the system of conservation laws, as well as four examples, considering all the four possible connections, namely, 1-shock/2-shock, 1-rarefaction/2-rarefaction, 1-rarefaction/2-shock and 1-shock/2-rarefaction are presented. [ABSTRACT FROM AUTHOR]

We solve exactly the Riemann problem for the non-linear hyperbolic system governing blood flow in human veins and note that, as modeled here, veins do not admit complete collapse, that is zero cross-sectional area A . This means that the Cauchy problem will not admit zero cross-sectional areas as initial condition. In particular, rarefactions and shock waves (elastic jumps), classical waves in the conventional Riemann problem, cannot be connected to the zero state with A = 0 . Moreover, we show that the area A * between two rarefaction waves in the solution of the Riemann problem can never attain the value zero, unless the data velocity difference u R − u L tends to infinity. This is in sharp contrast to analogous systems such as blood flow in arteries, gas dynamics and shallow water flows, all of which admitting a vacuum state. We discuss the implications of these findings in the modelling of the human circulation system that includes the venous system. [ABSTRACT FROM AUTHOR]

Recently there has been interest in studying a new class of elastic materials, which is described by implicit constitutive relations. Under some basic assumption for elasticity constants, the system of governing equations of motion for this elastic material is strictly hyperbolic but without the convexity property. In this paper, all wave patterns for the nonclassic nonlinearly elastic materials under Riemann data are established completely by separating the phase plane into twelve disjoint regions and by using a nonnegative dissipation rate assumption and the maximally dissipative kinetics at any stress discontinuity. Depending on the initial data, a variety of wave patterns can arise, and in particular there exist composite waves composed of a rarefaction wave and a shock wave. The solutions for a physically realizable case are presented in detail, which may be used to test whether the material belongs to the class of classical elastic bodies or the one wherein the stretch is expressed as a function of the stress. [ABSTRACT FROM AUTHOR]

A new simplified analytical model to analyze the penetration of rigid projectiles into soil media is presented. The soil medium is represented by a set of discs, responding in the radial direction under plain strain conditions. A convenient mathematical formulation is derived based on some simplifying assumptions. According to the present approach, the contact parameters in each disc are computed using the developed exact solution of the symmetrical Riemann problem for an irreversible compressible medium. One of the new features of the present model is the incorporation of the exact nonlinear equation of state including unloading-reloading thus considering another key variable that is the maximum medium density that is attained in the process of active loading before unloading is started. Thus the new model considers unloading in the soil medium during the progress of penetration. The present model focuses on the projectile motion and provides information on its velocity, deceleration and depth time histories. It also provides information on the interaction of the projectile with the surrounding soil such as the normal stress distribution along the projectile nose and is capable of determining the contact zone between the nose and the soil. Comparison of the present model results with two-dimensional numerical results as well as with different analytical models and with experimental data is performed. The present model predictions are found to be in good agreement with test data and superior to many existing simplified models. Contrary to many other simplified models, the present model is purely theoretical and does not require any empirical constants or special arbitrary assumptions for calculation of the contact pressures along the projectile nose at all times. The calculations require very small computer time and provide much information regarding the projectile motion in the soil medium. [ABSTRACT FROM AUTHOR]