Riemann Solutions Research Articles

Exact Riemann solutions of the Ripa model for flat and non-flat bottom topographies

Abstract This article is concerned with the derivation of exact Riemann solutions for Ripa model considering flat and non-flat bottom topographies. The Ripa model is a system of shallow water equations accounting for horizontal temperature gradients. In the case of non-flat bottom topography, the mass, momentum and energy conservation principles are utilized to relate the left and right states across the step-type bottom topography. The resulting system of algebraic equations is solved iteratively. Different numerical case studies of physical interest are considered. The solutions obtained from developed exact Riemann solvers are compared with the approximate solutions of central upwind scheme.

Three-dimensional discrete Boltzmann models for compressible flows in and out of equilibrium

We present a series of three-dimensional discrete Boltzmann models for compressible flows in and out of equilibrium. The key formulating technique is the construction of discrete equilibrium distribution function through inversely solving the kinetic moment relations that it satisfies. The crucial physical requirement is that all the used kinetic moment relations must be consistent with the non-equilibrium statistical mechanics. The necessity of such a kinetic model is that, with increasing the complexity of flows, the dynamical characterization of non-equilibrium state and the understanding of the constitutive relations need higher order kinetic moments and their evolution. The discrete Boltzmann models at the Euler and Navier–Stokes levels proposed by this scheme are validated by several well-known benchmarks, ranging from one-dimension to three-dimension. Particularly, when the local Mach number, temperature ratio, and pressure ratio are as large as 102, 104, and 105, respectively, the simulation results are still in excellent agreement with the Riemann solutions. How to model deeper thermodynamic non-equilibrium flows by discrete Boltzmann is indicated. Via the discrete Boltzmann method, it is convenient to simulate nonequilibrium flows without knowing exact form of the hydrodynamic equations.

Riemann solution for a class of morphodynamic shallow water dam-break problems

This paper investigates a family of dam-break problems over an erodible bed. The hydrodynamics is described by the shallow water equations, and the bed change by a sediment-conservation equation, coupled to the hydrodynamics by a sediment transport (bed-load) law. When the initial states $\boldsymbol{U}_{l}$ and $\boldsymbol{U}_{r}$ are sufficiently close to each other the resulting solutions are consistent with the theory proposed by Lax (Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, 1973, SIAM), that for a Riemann problem of $n$ equations there are $n$ waves associated with the $n$ characteristic families. However, for wet–dry dam-break problems over a mobile bed, there are three governing equations, but only two waves. One wave vanishes because of the presence of the dry bed. When initial left and right bed levels ($B_{l}$ and $B_{r}$) are far apart, it is shown that a semi-characteristic shock may occur, which happens because, unlike in shallow water flow on a fixed bed, the flux function is non-convex. In these circumstances it is shown that it is necessary to reconsider the usual shock conditions. Instead, we propose an implied internal shock structure the concept of which originates from the fact that the stationary shock over a fixed-bed discontinuity can be regarded as a limiting case of flow over a sloping fixed bed. The Needham & Hey (Phil. Trans. R. Soc. Lond. A, vol. 334, 1991, pp. 25–53) approximation for the ambiguous integral term $\int \!h\,\text{d}B$ in the shock condition is improved based on this internal shock structure, such that mathematically valid solutions that incorporate a morphodynamic semi-characteristic shock are arrived at.

Wave Interactions and Stability of Riemann Solutions of the Aw–Rascle Model for Generalized Chaplygin Gas

In this paper, we mainly study the interactions of the elementary waves of the Aw–Rascle model for the generalized Chaplygin gas. With the characteristic analysis method, the solutions are obtained constructively when the initial data are three piecewise constant states. Moreover, by studying the limits of the perturbed Riemann solutions as the perturbed parameter $\varepsilon$ tends to zero, we find that the Riemann solutions are stable for such perturbations on the initial data.

Numerical Study of Hydrodynamic and Solute Transport with Discontinuous Flows in Coastal Water

This paper aimed to develop a depth-averaged explicit model for flow and pollutant transport in coastal waters based on the shallow water equations and the mass advection-diffusion equation. The proposed model was discretized using the finite volume method (FVM) with triangular cells. Then, it applied Roe’s approximate Riemann solution to compute the water momentum flux on the grid interfaces. This model enabled the higher accuracy in capturing the dry-wet moving fronts (discontinuous problems for flow and solute). The high-resolution scheme was evaluated to solve the advection and diffusion terms for mass transport. The model was verified by comparing the predictions of analytical solutions, laboratory tests, and other simulations for Gironde estuary with good computational accuracy. The developed model was also used to calculate the circulation and the motion of chemical oxygen demand (COD) pollutants from the sewage outfalls in the Zhuanghe coastal water with dry and wet moving boundaries. The research results showed that the residual current directions of spring and neap tides were basically the same in the Zhuanghe coastal water. However, the tide residual current of spring tide was slightly greater than that of the neap tide. In addition, there were tide residual currents from the northeast to the southwest in nearshore water and from the southwest to the northeast outside the banks, respectively. The tidal flows in the alongshore direction were strong, resulting in highly spread concentration distributions. In particular, the COD concentration reached some parts of the southern water. It could be seen that the excessive pollutant discharge from the sewage outfalls located at Zhuanghe district would cause serious pollution in aquaculture water near Shicheng island.

Cone-grid scheme for solving hyperbolic systems of conservation laws and one application

In this work, we introduce a new numerical method, the so-called cone-grid scheme for solving hyperbolic systems of conservation laws. The formulation and the theoretical analysis for this scheme are slightly simpler than other algorithms. Simultaneously, our technique leads to a more efficacious numerical scheme. The Riemann solution is the basic ingredient for this scheme. We also give the main application of this technique, namely the global weak solutions for the one-dimensional model prescribes heat conduction in solids at low temperature, which called phonon–Bose model. This system consists of a conservation equation for the energy density and the heat flux. The scheme also satisfies positivity. The cone-grid scheme was compared with exact solution by three numerical examples, where explicit solutions are known. These numerical results verify the accuracy of the proposed scheme qualitatively and quantitatively.

Delta shock wave solution for a symmetric Keyfitz–Kranzer system

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.

An exact Riemann solver for wave propagation in arbitrary anisotropic elastic media with fluid coupling

We present a nonconformal mesh discontinuous Galerkin pseudospectral time domain algorithm for arbitrary anisotropic elastic/acoustic wave propagation problems. An exact Riemann solver is compactly derived to resolve the accurate coupling of multiple domains in the discontinuous Galerkin framework, including heterogeneous anisotropic solid–solid, acoustic–acoustic, and anisotropic solid–fluid interactions. We simplify the eigenvalue problem in the Riemann solution from the rank of 9 to 3, and introduce the generalized wave impedance with more physical insight. Validations and verifications with independent codes and analytical solutions illustrate the accuracy, flexibility, and stability of our algorithm.

The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas

This paper is concerned with the Euler equations in the magnetogasdynamics for generalized Chaplygin gas. The global solutions to the Riemann problems of the Euler equations in the magnetogasdynamics for generalized Chaplygin gas are obtained constructively by using phase plane analysis method. The formation of delta shock wave is studied as magnetic field vanishes. The limit behaviors of the Riemann solutions as magnetic field vanishes are also obtained.

Effect of random defects on dynamic fracture in quasi-brittle materials

We propose an asynchronous spacetime discontinuous Galerkin (aSDG) method combined with a novel rate-dependent interfacial damage model as a means to simulate crack nucleation and propagation in quasi-brittle materials. Damage acts in the new model to smoothly transition the aSDG jump conditions on fracture surfaces between Riemann solutions for bonded and debonded conditions. We use the aSDG method’s powerful adaptive meshing capabilities to ensure solution accuracy without resorting to crack-tip enrichment functions and extend those capabilities to support fracture nucleation, extension and intersection. Precise alignment of inter-element boundaries with flaw orientations and crack-propagation directions ensures mesh-independent crack-path predictions. We demonstrate these capabilities in a study of crack-path convergence as adaptive error tolerances tend to zero. The fracture response of quasi-brittle materials is highly sensitive to the presence and properties of microstructural defects. We propose two approaches to model these inhomogeneities. In the first, we represent defects explicitly as crack-like features in the analysis domain’s geometry with random distributions of size, location, and orientation. In the second, we model microscopic flaws implicitly, with probabilistic distributions of strength and orientation, to drive nucleation of macroscopic fractures. Crack-path oscillation, microcracking, and crack branching make numerical simulation of dynamic fracture particularly challenging. We present numerical examples that explore the influence of model parameters and inhomogeneities on fracture patterns and the aSDG model’s ability to capture complex fracture patterns and interactions.

On the Shallow Water Equations

AbstractWe studied the shallow water equations of nonlinear conservation laws. First we studied the parametrisation of nonlinear elementary waves and hence we present the solution to the Riemann problem. We also prove the uniqueness of the Riemann solution. The Riemann invariants are formulated. Moreover we give an interesting application of the Riemann invariants. We present the shallow water system in a diagonal form, which admits the existence of a global smooth solution for these equations. The other application is to introduce new conservation laws for the shallow water equations.

Zero dissipation limit to a Riemann solution for the compressible Navier-Stokes system of general gas

AbstractFor the general gas including ideal polytropic gas, we study the zero dissipation limit problem of the full 1-D compressible Navier-Stokes equations toward the superposition of contact discontinuity and two rarefaction waves. In the case of both smooth and Riemann initial data, we show that if the solutions to the corresponding Euler system consist of the composite wave of two rarefaction wave and contact discontinuity, then there exist solutions to Navier-Stokes equations which converge to the Riemman solutions away from the initial layer with a decay rate in any fixed time interval as the viscosity and the heat-conductivity coefficients tend to zero. The proof is based on scaling arguments, the construction of the approximate profiles and delicate energy estimates. Notice that we have no need to restrict the strengths of the contact discontinuity and rarefaction waves to be small.

The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws

The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws is considered. By the properties of wave curves, we observe that this system does not contain the composite wave compared to the barotropic models of gas dynamics with different pressure laws. Under some initial value data, the Riemann solution is constructed. Using the interaction of the elementary waves, we consider the generalized Riemann problem and discover that the Riemann solution is stable for such perturbation of the initial data.

A random sampling method for a family of Temple-class systems of conservation laws

The Aw–Rascle–Zhang traffic model, a model of sedimentation, and other applications lead to nonlinear $$2 \times 2$$ systems of conservation laws that are governed by a single scalar system velocity. Such systems are of the Temple class since rarefaction wave curves and Hugoniot curves coincide. Moreover, one characteristic field is genuinely nonlinear almost everywhere, and the other is linearly degenerate. Two well-known problems associated with these systems are handled via a random sampling approach. Firstly, Godunov’s and related methods produce spurious oscillations near contact discontinuities since the numerical solution invariably leaves the invariant region of the exact solution. It is shown that alternating between averaging (Av) and remap steps similar to the approach by Chalons and Goatin (Commun Math Sci 5:533–551, 2007) generates numerical solutions that do satisfy an invariant region property. If the remap step is made by random sampling (RS), then combined techniques due to Glimm (Commun Pure Appl Math 18:697–715, 1965), LeVeque and Temple (Trans Am Math Soc 288:115–123, 1985) prove that the resulting Av–RS scheme converges to a weak solution. Numerical examples illustrate that the new scheme is superior to Godunov’s method in accuracy and resolution. Secondly, the vacuum state, which may form even from positive initial data, causes potential problems of non-uniqueness and instability. This is resolved by introducing an alternative Riemann solution concept.

Non-selfsimilar global solutions and their structure for the multi-dimensional combustion models

We investigate the Chapman-Jouguet model in multi-dimensional space, and construct explicitly its non-selfsimilar Riemann solutions. By the method we apply in this paper, general initial discontinuities can be dealt with, even for complex interaction of combustion waves. Furthermore, we analyze the way in which the area of unburnt gas shrinks.

SWE-SPHysics Simulation of Dam Break Flows at South-Gate Gorges Reservoir

This paper applied a Smoothed Particle Hydrodynamics (SPH) approach to solve Shallow Water Equations (SWEs) to study practical dam-break flows. The computational program is based on the open source code SWE-SPHysics, where a Monotone Upstream-centered Scheme for Conservation Laws (MUSCL) reconstruction method is used to improve the Riemann solution with Lax-Friedrichs flux. A virtual boundary particle method is applied to treat the solid boundary. The model is first tested on two benchmark collapses of water columns with the existence of downstream obstacle. Subsequently the model is applied to forecast a prototype dam-break flood, which might occur in South-Gate Gorges Reservoir area of Qinghai Province, China. It shows that the SWE-SPH modeling approach could provide a promising simulation tool for practical dam-break flows in engineering scale.

The limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term

The paper is concerned with the limit behavior of the Riemann solutions to the inhomogeneous generalized Chaplygin gas equations. The formation of delta shock waves and the vacuum states are identified and analyzed as the pressure vanishes. Unlike the homogeneous case, the Riemann solutions are no longer self-similar. As the pressure vanishes, the Riemann solutions to the generalized Chaplygin gas equations with a friction term converge to the Riemann solutions to the pressureless gas dynamics model with a body force.

The invariant region for the special gas dynamics system

In this paper, we study the invariant region for the special gas dynamics system, where the conservation law of energy in classical Euler equations is replaced by the specific entropy. Firstly, the Riemann solutions for five domains in the u–p plane are completely constructed. Then we obtain the bounded convex invariant region of the Riemann solutions through analyzing structure of the Riemann solutions. Moreover, with the bounded convex invariant region, we get the uniform boundedness of approximate solutions, which is the foundation to prove the existence of weak solutions for the special gas dynamic system using compensated compactness method.

On periodic wave solutions of the KdV6 equation via bilinear Bäcklund transformation

The KdV6 equation is an integrable soliton equation. By using the bilinear Bäcklund transformation, we derive the one and two Riemann theta function periodic wave solutions of the KdV6 equation. The dynamical behaviors of the periodic wave solutions are analyzed by numerical simulations. Finally, we establish the relations between the periodic wave solutions and the soliton solutions.

Approximation of generalized Riemann solutions to compressible Euler-Poisson equations of isothermal flows in spherically symmetric space-times

In this paper, we consider the compressible Euler-Poisson system in spherically symmetric space-times. This system, which describes the conservation of mass and momentum of physical quantity with attracting gravitational potential, can be written as a $3\times 3$ mixed-system of partial differential systems or a $2\times 2$ hyperbolic system of balance laws with $global$ source. We show that, by the equation for the conservation of mass, Euler-Poisson equations can be transformed into a standard $3\times 3$ hyperbolic system of balance laws with $local$ source. The generalized approximate solutions to the Riemann problem of Euler-Poisson equations, which is the building block of generalized Glimm scheme for solving initial-boundary value problems, are provided as the superposition of Lax's type weak solutions of the associated homogeneous conservation laws and the perturbation terms solved by the linearized hyperbolic system with coefficients depending on such Lax solutions.