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In this work, we explore the Riemann problem of the Fokas–Lenells (FL) equation given initial data in the form of a step discontinuity by employing the Whitham modulation theory. The periodic wave solutions of the FL equation are characterized by elliptic functions along with the Whitham modulation equations. Moreover, we find that the ±$\pm$ signs for the velocities of the periodic wave solutions remain unchanged during propagation. Thus, when analyzing the propagation behavior of solutions, it is necessary to separately consider the clockwise (negative velocity) and counterclockwise (positive velocity) cases. In this regard, we present the classification of the solutions to the Riemann problem of the FL equation in both clockwise and counterclockwise cases for the first time. [ABSTRACT FROM AUTHOR]

The Riemann problem for the discrete conservation law 2u̇n+un+12−un−12=0$2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a quasi‐continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogs of well‐known dispersive hydrodynamic solutions—rarefaction waves (RWs) and dispersive shock waves (DSWs)—additional unsteady solution families and finite‐time blowup are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counterpropagating periodic wavetrains with the same frequency connected to a partial DSW or an RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations. [ABSTRACT FROM AUTHOR]

In a gas–water interaction problem, the nonlinear relationship between shock wave velocity is introduced into a Hugoniot curve, and a Mie–Grüneisen Equation of state (EOS) is established by setting the Hugoiot curve as the reference state. Unlike other simple EOS based on the thermodynamics laws of gas (such as the Tait EOS), the Mie–Grüneisen EOS uses reference states to cover an adiabatic impact relationship and considers the thermodynamics law separately. However, the expression of the EOS becomes complex, and it is not adaptive to many methods. A multicomponent Mie–Grüneisen mixture model is employed in this study to conquer the difficulty of the complex form of an EOS. In this model, some coefficients in the Mie–Grüneisen EOS are regarded as variables and solved using newly constructed equations. The performance of the Mie–Grüneisen mixture model in the gas–water problem is tested by low-compression cases and high-compression cases. According to these two tests, it is found that the numerical solutions of the shock wave under the Mie–Grüneisen EOS agrees with empirical data. When compared to other simple-form EOSs, it is seen that the Mie–Grüneisen EOS has slight advantages in the low-compression case, but it plays an important role in the high-compression case. The comparison results show that the solution of the simple-form EOS clearly disagrees with the empirical data. A further study shows that the gap between the Mie–Grüneisen EOS and other simple-form EOSs becomes larger as the initial pressure and particle velocity increase. The impact effects on the pressure, density and particle velocity are studied. Moreover, the gas–water interaction in a spherical coordinate plane and a two-dimensional coordinate is a significant part of our work. [ABSTRACT FROM AUTHOR]

Condensation induces both heat addition and mass consumption effects on the carrier flow field, leading to complex wave patterns. In this study, we aim to elucidate the structures of flow field under the influence of condensation. To achieve this, we present a mathematical model, which is formulated using Euler equations with a singular source term. Given the difficulties posed by both weak solution theory and the Dal Maso–LeFloch–Murat (DLM) theory (J. Math. Pures Appl. 483‐548 (1995)) in defining solutions to the Riemann problem, we propose a novel approach. This involves the coupling of weak solutions within two subregions, taking into consideration the discontinuity of source term. Firstly, we identify two stationary discontinuities originating from the singular source term—namely, stationary waves and composite waves. Admissibility criterions are developed for these discontinuities to facilitate the selection of physically meaningful solutions. Secondly, we employ a double classical Riemann problems (CRPs) framework to analyze the structures of Riemann solution. Our analysis reveals that the Riemann solution may exhibit seven structures with stationary wave and four structures with composite wave. Finally, the proposed model and its theoritical results are applied to validate the structure of the flow field with condensation discontinuities. For the pure heat addition problem, we successfully demonstrate all wave patterns of the flow field, perfectly matching the outcomes of numerical experiments. [ABSTRACT FROM AUTHOR]

For real-world engineering applications, there is a great deal of interest in developing effective models and simulations of continuum-rarefied gas flows. In this study, the numerical simulations of Grad's 13 (G13) moment equations with application to the Riemann problem in a wide range of continuum-rarefied flow regimes are presented. This work emphasizes numerical robustness and wave phenomena in the G13 system to provide a building block for regularized 13 moment systems, and high-order Grad's models. For this purpose, a high-order modal discontinuous Galerkin solver is developed for solving one-dimensional G13 moment equations. For spatial discretization, hierarchical modal basis functions premised on orthogonal-scaled Legendre polynomials are used. The proposed approach reduces the G13 systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order SSP Runge-Kutta algorithm. Three Riemann test cases, including the shock tube, two shock waves and two rarefaction waves, are examined in continuum-rarefied flow regimes. In the G13 system, the arising characteristic waves and dissipation phenomena are investigated in depth. The numerical results demonstrate that every Riemann problem does not have a solution for the G13 system because of loss of hyperbolicity of the system. [ABSTRACT FROM AUTHOR]

The problems of constructing an exact or approximate solution of system of singular integro‐differential equations related to the problems of adhesive interaction between elastic thin finite or infinite nonhomogeneous patch and elastic plate are investigated. For the patch loaded with horizontal and vertical forces, the usual model of beam bending in combination with the uniaxial stress state model is valid. Using the methods of theory of analytic functions, integral transformation, or orthogonal polynomials, the singular integro‐differential equations reduced to the different boundary value problems (Karleman type problem with displacements and Riemann problem) of the theory of analytic functions or to the infinite system of linear algebraic equations. The asymptotic analysis of problem is carried out. [ABSTRACT FROM AUTHOR]

This paper studies constrained Newtonian fluid flows through porous media, accounting for the drag effect on the fluid, modeled using a Mixture Theory perspective and a constitutive relation for the pressure—namely, a continuous and differentiable function of the saturation that ensures always preserving the problem hyperbolicity. The pressure equation also permits an ultra-small porous matrix supersaturation (that is controlled) and the transition from unsaturated to saturated flow (and vice versa). The mathematical model gives rise to a nonlinear, non-homogeneous hyperbolic system. Its numerical simulation combines Glimm's method with an operator-splitting strategy to account for the Darcy and Forchheimer terms that cause the system's non-homogeneity. Despite the Glimm method's proven convergence, it is not adequate to approximate non-homogeneous hyperbolic systems unless combined with an operator-splitting technique. Although other approaches have already addressed this problem, the novelty is combining Glimm's method with operator-splitting to account for linear and nonlinear drag effects. Glimm's scheme marches in time using a formerly selected number of associated Riemann problems. The constitutive relation for the pressure—an increasing function of the saturation, with the first derivative also increasing, convex, and positive, enables us to obtain explicit expressions for the Riemann invariants. The results show the influence of the Darcy and Forchheimer drag terms on the flow. [ABSTRACT FROM AUTHOR]

Using the spectral transform on a torus, we solve the initial–boundary value problem for quasi-one-dimensional excitations in a semibounded ferromagnet, taking the exchange interaction, orthorhombic anisotropy, and magnetostatic fields into account. We used the mixed boundary conditions whose limit cases correspond to free and fully pinned spins at the sample edge. We predict and analyze new types of solitons (moving domain walls and precessing breathers), whose cores are strongly deformed near the sample boundary. At large distances from the sample surface, they take the form of typical solitons in an unbounded medium. We analyze the properties of the reflection of solitons from the sample boundary depending on the degree of spin pinning at the surface. We obtain new conservation laws that guarantee the true boundary conditions to hold when solitons reflect from the sample surface. [ABSTRACT FROM AUTHOR]

We investigate the wave structure and new phenomena of the Riemann problems of isentropic compressible Euler equations with discontinuous flux in momentum caused by different equations of states, including pressureless flow and Chaplygin gas. Specifically, we focus on solutions within the class of Radon measures. To resolve the discontinuous flux, we introduce a delta shock that admits mass concentration between the pressureless flow on the left and Chaplygin gas on the right. By exploring both the classical and singular Riemann problems, we find that a global delta shock solution exists, satisfying the over-compressing condition. This finding is a generalization of classical theories on Riemann problems. In particular, we demonstrate that a vacuum left state and right Chaplygin gas can always be connected by a global delta shock satisfying the over-compressing condition. For singular Riemann problems, influenced by initial velocity, we observe that for some initial data, the composite wave comprises contact discontinuities, vacuum, and a local delta shock satisfying the over-compressing condition. Through a detailed analysis of the intricate interactions between contact discontinuities and delta shocks, we show that this local solution can be extended globally. [ABSTRACT FROM AUTHOR]

The present study is concerned with the analytical solution of Riemann problem for conservation laws of van der Waals gas. By utilizing Rankine–Hugoniot conditions and Lax entropy condition, we derive classical wave solution of Riemann problem and analyze their properties. Also, it is observed here that van der Waals gasdynamics system is more complex in comparison to ideal gasdynamics case. Further, the effect of presence of intermolecular forces of attraction between the particles and variation of covolume of the gas on the density and velocity distribution across the simple wave, shock wave and contact discontinuities is discussed. [ABSTRACT FROM AUTHOR]

We are concerned with the Riemann problem for the isentropic Euler equations of mixed type in the dark energy fluid. This system is non-strictly hyperbolic on the boundary curve of elliptic and hyperbolic regions. We obtain the unique admissible shock waves by utilizing the viscosity criterion. Assuming fixed left states are in the elliptic and hyperbolic regions, respectively, we construct the unique Riemann solution for the mixed-type models with the initial right state in some feasible regions. Finally, we present numerical simulations which are consistent with our theoretical results. [ABSTRACT FROM AUTHOR]

The 3 × 3 blood flow dynamic model describes the flow of blood in flexible vessels. We study the inviscous blood flow in arteries model in this paper. The elementary waves of the blood flow in arteries include the rarefaction wave, the shock wave and the stationary wave which appears where the material properties of vessel wall change. The interactions of stationary wave with rarefaction wave and shock wave in arteries are discussed in detail. We focus on the changes of the cross-sectional area of the blood vessel and the averaged axial velocity of blood flow after the rarefaction wave and the shock wave penetrate the stationary wave. They change after interactions. [ABSTRACT FROM AUTHOR]

The stability for magnetic field to the solution of the Riemann problem for the polytropic fluid in a variable cross-section duct is discussed. By the vanishing magnetic field method, the stable solutions are determined by comparing the limit solutions with the solutions of the Riemann problem for the polytropic fluid in a duct obtained by the entropy rate admissibility criterion. [ABSTRACT FROM AUTHOR]

Undercompressive shocks and their role in solving Riemann problem are studied. Solutions to a special system of two hyperbolic equations representing conservation laws are investigated. On the one hand, this system of equations makes it possible to demonstrate the nonstandard solutions to the Riemann problem; on the other hand, this system of equations describes longitudinal–torsional waves in elastic rods. We use the traveling wave criterion for admissibility of shocks as the additional jump condition. If the dissipation parameters included in each of the equations of the system are different, then there are undercompressed waves. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Riemann problem and wave interactions for a 2 × 2 non-strictly hyperbolic system of conservation laws with non-convex flux. Riemann problem can not be solved always in the usual class of elementary waves and the solution develops δ -waves. We construct the exact solution to the Riemann problem for the governing system and discuss all possible cases of wave interactions involving δ -wave. [ABSTRACT FROM AUTHOR]

When an impactor strikes a layered target, both the impactor and the target experience waves. The waves produced travel and engage in interactions with other waves as well as the interfaces in the impactor-target system. For the impact problems on a layered mediumwith periodic properties and layered elastic media of Goupillaud-type (each layer has the same wave travel time), researchers have presented an analytical solution for stress variation with position and time within the target. However, the solution for an elastic media not satisfying the above conditions is not available in the literature. The present study fills this gap and finds the behaviour of a generalized layered medium to an impact problem. The response of the material at any position inside the layered medium is found by solving the interaction between waves, interfaces, and boundaries. The mass, momentum balance and constitutive relationship are solved to get the exact analytical expressions for particle velocity and stress for each possible wave interaction happening in the impactor and the layered medium. The expressions are utilized in a computer program to study the impact behaviour of a layered media. The code tracks each wave as it travels through the system and identifies those interactions that occur in the shortest time, uses the stress and velocity expression for that interaction, and updates the state of the material. When stress produced at the impact surface is tensile in nature, the impactor and target can be separated. The work can be applied to both finite and semi-infinite impactors and targets, and the layered medium does not necessarily have to be a periodic layered media or a Goupillaud-type medium. [ABSTRACT FROM AUTHOR]

In this paper, we obtained the solutions of the Riemann problem for a quasilinear hyperbolic system with four equations characterizing one-dimensional planar and radially symmetric flow of van der Waals reacting gases with dust particles involving shock wave, simple wave and contact discontinuities without any restriction on the magnitude of initial states. This system is more complex due to the dust particles in van der Waals reacting gases, that is, typical irreversible exothermic reaction of real gases in the presence of dust particles. The generalised Riemann invariants are used to determine the necessary and sufficient condition for the uniqueness of solutions. The effects of non-idealness and dust particles on the compressive and rarefaction waves are also analyzed. [ABSTRACT FROM AUTHOR]

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]

Motivated by the foam displacement in porous media with linear adsorption, we extended the existing framework for two-phase flow containing an active tracer described by a non-- strictly hyperbolic system of conservation laws. We solved the global Riemann problem by presenting possible wave sequences that composed this solution. Although the problems are well-posed for all Riemann data, there is a parameter region where the solution lacks structural stability. We verified that the model implemented on the most used commercial solver for geoscience, CMG-STARS, describing foam displacement in porous media with adsorption, satisfies the hypotheses to apply the developed theory, resulting in structural stability loss for some parameter regions. [ABSTRACT FROM AUTHOR]

The ignition problem for the Chaplygin system is considered. Under the entropy conditions, we obtain constructively the unique solution and discover that the combustion wave solutions may be extinguished for some cases. Especially, we obtain that the combustion wave occurs although there is no combustion before. The transition between deflagration and detonation is also shown. [ABSTRACT FROM AUTHOR]

The theory of small-amplitude waves propagating across a blood vessel junction has been well established with linear analysis. In this study, we consider the propagation of large-amplitude, nonlinear waves (i.e. shocks and rarefactions) through a junction from a parent vessel into two (identical) daughter vessels using a combination of three approaches: numerical computations using a Godunov method with patching across the junction, analysis of a nonlinear Riemann problem in the neighbourhood of the junction and an analytical theory which extends the linear analysis to the following order in amplitude. A unified picture emerges: an abrupt (prescribed) increase in pressure at the inlet to the parent vessel generates a propagating shock wave along the parent vessel which interacts with the junction. For modest driving, this shock wave divides into propagating shock waves along the two daughter vessels and reflects a rarefaction wave back towards the inlet. However, for larger driving the reflected rarefaction wave becomes transcritical, generating an additional shock wave. Just beyond criticality this new shock wave has zero speed, pinned to the junction, but for further increases in driving this additional shock divides into two new propagating shock waves in the daughter vessels.

In a gas–water interaction problem, the nonlinear relationship between shock wave velocity is introduced into a Hugoniot curve, and a Mie–Grüneisen Equation of state (EOS) is established by setting the Hugoiot curve as the reference state. Unlike other simple EOS based on the thermodynamics laws of gas (such as the Tait EOS), the Mie–Grüneisen EOS uses reference states to cover an adiabatic impact relationship and considers the thermodynamics law separately. However, the expression of the EOS becomes complex, and it is not adaptive to many methods. A multicomponent Mie–Grüneisen mixture model is employed in this study to conquer the difficulty of the complex form of an EOS. In this model, some coefficients in the Mie–Grüneisen EOS are regarded as variables and solved using newly constructed equations. The performance of the Mie–Grüneisen mixture model in the gas–water problem is tested by low-compression cases and high-compression cases. According to these two tests, it is found that the numerical solutions of the shock wave under the Mie–Grüneisen EOS agrees with empirical data. When compared to other simple-form EOSs, it is seen that the Mie–Grüneisen EOS has slight advantages in the low-compression case, but it plays an important role in the high-compression case. The comparison results show that the solution of the simple-form EOS clearly disagrees with the empirical data. A further study shows that the gap between the Mie–Grüneisen EOS and other simple-form EOSs becomes larger as the initial pressure and particle velocity increase. The impact effects on the pressure, density and particle velocity are studied. Moreover, the gas–water interaction in a spherical coordinate plane and a two-dimensional coordinate is a significant part of our work.

This paper is concerned with Riemann problem for a class of non-strictly hyperbolic systems of conservation laws. The two kinds of Riemann solutions including vacuum and delta shock wave are constructed. Under the generalized Rankine-Hugoniot relation and entropy condition, we establish the existence and uniqueness of delta shock wave solutions. Furthermore, by studying the interactions among of the delta shock wave and vacuum as well as contact discontinuity, the Riemann solutions with four kinds of different structures are obtained. Additionally, the stability of the Riemann solutions is obtained under certain perturbation of the initial data. [ABSTRACT FROM AUTHOR]

The complete set of exact solutions to the Riemann problem for a novel second-order continuous macroscopic traffic flow model proposed by Hwang and Yu (J Comput Phys 350:927–950, 2017) is constructively solved in explicit forms by choosing the specific driving function of momentum. Especially, a hyperbolic composite wave is found in certain Riemann solution under the suitable initial condition, where a delta contact discontinuity is attached on the head of a rarefaction wave. Moreover, the asymptotic behavior of different Riemann solutions is explored and discussed, respectively, to analyze the influence of the two perturbed parameters in this model comprehensively. Additionally, the offered numerical experiments are well identical with our theoretic results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the Riemann problem for two‐layered blood flow model, which is represented by a system of quasi‐linear hyperbolic partial differential equations (PDEs) derived from the Euler equations by vertical averaging across each layer. We consider the Riemann problem with varying velocities and equal constant density through arteries. For instance, the flow layer close to the wall of vessel has a slower average speed than the layer far from the vessel because of the viscous effect of the blood vessel. We first establish the existence and uniqueness of the corresponding Riemann solution by a thorough investigation of the properties of elementary waves, namely, shock wave, rarefaction wave, and contact discontinuity wave. Further, we extensively analyze the elementary wave interaction between rarefaction wave and shock wave with contact discontinuity and rarefaction wave and shock wave. The global structure of the Riemann solutions after each wave interaction is explicitly constructed and graphically illustrated towards the end. [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]

The step-like initial value problem of the Fokas–Lenells equation is discussed based on Whitham modulation theory. Via the finite-gap integration method, the zero-phase, one-phase, N-phase solutions, and corresponding Whitham equation are obtained. Analytical and graphical methods are used to provide elementary wave structures of rarefaction waves and dispersive shock waves, which allows the classification of all wave structures that evolve from initial discontinuities. Finally, two typical Riemann problems are solved and represented graphically. [ABSTRACT FROM AUTHOR]

This article discusses the Riemann problem for the Chaplygin gas Euler equations that include the presence of two source terms. By means of variable substitution, two kinds of non-self-similar Riemann solutions involving delta-shock are constructed explicitly. For the delta-shock, the generalized Rankine–Hugoniot relations and the over-compressive entropy condition are clarified. Moreover, the position, propagation speed and strength of the delta-shock are given explicitly. It is discovered that the position of the delta-shock is a combination of an exponential function and a linear function, and the weight of the delta-shock is an exponential function of the time. Interestingly, even when the delta-shock is a straight line, the weight of the delta-shock is no longer a linear function of the time t. In addition, it is proved that the Riemann solutions converge to the corresponding ones of Chaplygin gas Euler equations with friction as k drops to zero, and the Riemann solutions converge to the corresponding ones of Chaplygin gas Euler equations as k and β tend to zero simultaneously. Furthermore, it is also shown that the limits of Riemann solutions are just the Riemann solutions to the transport equations with same source terms as the Chaplygin gas pressure falls to zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of hyperbolic systems of conservation laws admitting delta-waves. The Riemann problem is completely solved. The solutions exhibit four kinds of wave patterns: the first contains a vacuum state bounded by two contact discontinuities; the second involves the overcompressible delta-wave; the third and fourth include a vacuum state bounded by a contact delta-wave and a contact discontinuity. The highlight of this paper is the contact delta-waves. [ABSTRACT FROM AUTHOR]

The solution of a quasi-gas dynamic (QGD) system of equations using the local discontinuous Galerkin method (LDG) is considered. One-dimensional Riemann discontinuity problems with known exact solutions are solved. Strong discontinuities are present in the solutions of the problems. Therefore, to ensure the monotonicity of the solution obtained by the LDG method, the so-called slope limiters, or limiters, are introduced. A "moment" limiter is chosen that preserves as high an order as possible. The limiter is modified to smooth the oscillations in the areas where the solution is constant. [ABSTRACT FROM AUTHOR]

We consider a boundary value problem for a system of integro-differential equations arising when modeling the influence of an electric field on a plasma layer. The paper presents an analytical solution of this problem, which is constructed with the use of theory of the Fourier transform of generalized functions and the Gakhov–Muskhelishvili method for solving singular integral equations and the Riemann boundary value problem. [ABSTRACT FROM AUTHOR]

In the present paper, the authors consider the perturbed initial value problem of the Chapman-Jouguet model for the Chaplygin gas. We obtain the unique solution by analyzing the elementary waves under the global entropy conditions. We observe that the combustion wave solution may be extinguished after perturbation which tells the instability of the unburnt gas. And we also capture the transitions between the deflagration wave and the detonation wave. [ABSTRACT FROM AUTHOR]

The accurate Riemann solutions in fully explicit forms are obtained for the one-dimensional inviscid, compressible, isothermal liquid–gas two-phase flow model of drift-flux type under the action of the unique external force of gravity for a pipeline placed on an inclined plane. It is also shown that the Riemann solution involves either a curved delta shock wave or the vacuum state for the pressureless case. Moreover, the asymptotic limits of Riemann solutions are investigated in detail by letting the pressure drop to zero, in which the formation of curved delta shock wave is obtained from the curved Riemann solution made up of backward shock wave, middle contact discontinuity and forward shock as well as the formation of vacuum state is also achieved from the curved Riemann solution consisting of backward rarefaction wave, middle contact discontinuity and forward rarefaction wave. In addition, some representative numerical results are offered to confirm the formation of delta shock wave and vacuum state. [ABSTRACT FROM AUTHOR]

This paper presents fluid dynamics simulation data associated with two test cases in the related research article [1]. In this article, an efficient bimaterial Riemann problem solver is proposed to accelerate multi-material flow simulations that involve complex thermodynamic equations of state and strong discontinuities across material interfaces. The first test case is a one-dimensional benchmark problem, featuring large density jump (4 orders of magnitude) and drastically different thermodynamics relations across a material interface. The second test case simulates the nucleation of a pear-shaped vapor bubble induced by long-pulsed laser in water. This multiphysics simulation combines laser radiation, phase transition (vaporization), non-spherical bubble expansion, and the emission of acoustic and shock waves. Both test cases are performed using the M2C solver, which solves the three-dimensional Eulerian Navier-Stokes equations, utilizing the accelerated bimaterial Riemann solver. Source codes provided in this paper include the M2C solver and a standalone version of the accelerated Riemann problem solver. These source codes serve as references for researchers seeking to implement the acceleration algorithms introduced in the related research article. Simulation data provided include fluid pressure, velocity, density, laser radiance and bubble dynamics. The input files and the workflow to perform the simulations are also provided. These files, together with the source codes, allow researchers to replicate the simulation results presented in the research article, which can be a starting point for new research in laser-induced cavitation, bubble dynamics, and multiphase flow in general.

A stable scheme is proposed in this paper in order to obtain approximate solutions of second-moment turbulent models for incompressible flows with or without thermal transport equation. The analysis of the convective terms, which includes the solution of the associated Riemann problem, enables to propose a standard projection scheme, and to get rid of spurious oscillations.

This paper studies constrained Newtonian fluid flows through porous media, accounting for the drag effect on the fluid, modeled using a Mixture Theory perspective and a constitutive relation for the pressure—namely, a continuous and differentiable function of the saturation that ensures always preserving the problem hyperbolicity. The pressure equation also permits an ultra-small porous matrix supersaturation (that is controlled) and the transition from unsaturated to saturated flow (and vice versa). The mathematical model gives rise to a nonlinear, non-homogeneous hyperbolic system. Its numerical simulation combines Glimm’s method with an operator-splitting strategy to account for the Darcy and Forchheimer terms that cause the system’s non-homogeneity. Despite the Glimm method’s proven convergence, it is not adequate to approximate non-homogeneous hyperbolic systems unless combined with an operator-splitting technique. Although other approaches have already addressed this problem, the novelty is combining Glimm’s method with operator-splitting to account for linear and nonlinear drag effects. Glimm’s scheme marches in time using a formerly selected number of associated Riemann problems. The constitutive relation for the pressure—an increasing function of the saturation, with the first derivative also increasing, convex, and positive, enables us to obtain explicit expressions for the Riemann invariants. The results show the influence of the Darcy and Forchheimer drag terms on the flow.

We are concerned with the Riemann problem for the isentropic Euler equations of mixed type in the dark energy fluid. This system is non-strictly hyperbolic on the boundary curve of elliptic and hyperbolic regions. We obtain the unique admissible shock waves by utilizing the viscosity criterion. Assuming fixed left states are in the elliptic and hyperbolic regions, respectively, we construct the unique Riemann solution for the mixed-type models with the initial right state in some feasible regions. Finally, we present numerical simulations which are consistent with our theoretical results.

The Riemann problem for a pressureless hydrodynamic model is solved explicitly, whose solution is either a contact-vacuum-contact wave or a delta shock wave to connect the left and right constant states. Moreover, the global perturbed Riemann solutions for this model are also constructed by investigating carefully all the presentive delta shock interaction problems when the initial condition is made up of three piecewise constant states. Finally, the stability of Riemann solutions is also proved by letting the perturbed parameter tend to zero under this specially designated small perturbation of this step-like initial condition. [ABSTRACT FROM AUTHOR]

This work uses a mixture theory approach to describe kinematically constrained flows through porous media using an adequate constitutive relation for pressure that preserves the problem hyperbolicity even when the flow becomes saturated. This feature allows using the same mathematical tool for handling unsaturated and saturated flows. The mechanical model can represent the saturated–unsaturated transition and vice-versa. The constitutive relation for pressure is a continuous and differentiable function of saturation: an increasing function with a strictly convex, increasing, and positive first derivative. This significant characteristic permits the fluid to establish a tiny controlled supersaturation of the porous matrix. The associated Riemann problem's complete solution is addressed in detail, with explicit expressions for the Riemann invariants. Glimm's semi-analytical scheme advances from a given instant to a subsequent one, employing the associated Riemann problem solution for each two consecutive time steps. The simulations employ a variation in Glimm's scheme, which uses the mean of four independent sequences for each considered time, ensuring computational solutions with reliable positions of rarefaction and shock waves. The results permit verifying this significant characteristic. [ABSTRACT FROM AUTHOR]

The solutions of the Riemann problem for the isentropic Euler equations with the specific three independent constitutive pressure laws are obtained constructively for four different structures. The formation of delta shock wave and the combination between two contact discontinuities in the Riemann solutions for the isentropic Chaplygin Euler equations are explored in detail by letting the isothermal and logarithmic-corrected equation of state tend to zero and simultaneously remaining the Chaplygin gas equation of state unchanged in the total pressure term. In addition, the numerical experiments are carried out to confirm the delta shock formation. [ABSTRACT FROM AUTHOR]

In this article, we consider the Riemann problem with arbitrary initial data for one-dimensional blood flow equations in the arterial circulation. Here, we establish the existence of the self-similar solution to the Riemann problem by limiting viscosity approach. We convert the Riemann problem to a boundary value problem by adding a suitable viscosity term and establish the existence of the solution. Finally, we construct the existence of the solution to this Riemann problem in the presence of vacuum state. [ABSTRACT FROM AUTHOR]

In this article, we study the delta shock wave for zero-pressure gasdynamics system with energy conservation laws in the frame of α -solutions defined in the setting of distributional products. By reformulating the system, we construct within a convenient space of distributions, all solutions which include discontinuous solutions and Dirac delta measures. We also establish the generalized Rankine–Hugoniot jump conditions for delta shock waves. The α -solutions which we constructed coincide with the solution obtained through different methods. [ABSTRACT FROM AUTHOR]

It is interesting and challenging to study conservation laws with discontinuous source terms and explore how the delta shock wave is influenced by the discontinuous source term. However, so far, few results have been obtained about it. In this paper, the Riemann problem for the pressureless Euler equations with a discontinuous source term is considered. The delta shock wave solution is obtained by combining the generalized Rankine-Hugoniot conditions, together with the method of characteristics for various kinds of different situations, and the impact of the discontinuous source term on the delta shock front are precisely illustrated. Moreover, during the process of constructing the Riemann solution, some interesting phenomena are also observed, such as the disappearance of the delta shock wave and the occurrence of the vacuum state, etc. [ABSTRACT FROM AUTHOR]