The cell transmission model (CTM) is a macroscopic model that describes the dynamics of traffic flow over time and space. The effectiveness and accuracy of the CTM are discussed in this paper. First, the CTM formula is recognized as a finite-volume discretization of the kinematic traffic model with a trapezoidal flux function. To validate the constructed scheme, the simulation of shock waves and rarefaction waves as two important elements of traffic dynamics was performed. Adaptation of the CTM for intersecting and splitting cells is discussed. Its implementation on the road segment with traffic influx produces results that are consistent with the analytical solution of the kinematic model. Furthermore, a simulation on a simple road network shows the back and forth propagation of shock waves and rarefaction waves. Our numerical result agrees well with the existing result of Godunov’s finite-volume scheme. In addition, from this accurately proven scheme, we can extract information for the average travel time on a certain route, which is the most important information a traveller needs. It appears from simulations of different scenarios that, depending on the circumstances, a longer route may have a shorter travel time. Finally, there is a discussion on the possible application for traffic management in Indonesia during the Eid al-Fitr exodus.

The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we consider a ‘spherical piston’ problem for the relativistic Euler equations, which describes the wave motion produced by a sphere expanding into an infinite surrounding medium. We use the reflected characteristics method to construct a global piecewise smooth solution with a single shock of this spherical piston problem, provided that the speed of the sphere is a small perturbation of a constant speed.

The study of radially symmetric motion is important for the theory of explosion waves. We construct rigorously self-similar entropy solutions to Riemann initial-boundary value problems for the radially symmetric relativistic Euler equations. We use the assumption of self-similarity to reduce the relativistic Euler equations to a system of nonlinear ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. For the ultra-relativistic Euler equations, we also obtain the uniqueness of the self-similar entropy solution to the Riemann initial-boundary value problems.

The present paper concerns the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ, ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves, δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.

In this study, we propose a high order well-balanced weighted compact nonlinear (WCN) scheme for the gas dynamic equations under gravitational fields. The proposed scheme is an extension of the high order WCN schemes developed in (S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321). For the purpose of maintaining the exact steady state solution, the well-balanced technique in (Y. Xing, C.-W Shu, J. Sci. Comput. 54 (2013) 645-662) is employed to split the source term into two terms. The proposed scheme can maintain the isothermal equilibrium solution exactly, genuine high order accuracy and resolve small perturbations of the hydrostatic balance state on the coarse meshes. Furthermore, in order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact schemes with local characteristic projections are used to construct different WCN schemes. Several representative one- and two-dimensional examples are simulated to demonstrate the good performance of the proposed schemes.

In this study, a numerical framework of the high order well-balanced weighted compact nonlinear (WCN) schemes is proposed for the shallow water equations based on the work in [S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321]. We employ a special splitting technique for the source term proposed in [Y. Xing, C.-W Shu, J. Comput. Phys. 208 (2005) 206-227] to maintain the exact C-property, which can be proved theoretically. In the meantime, the genuine high order accuracy of the numerical scheme can be observed successfully, and small perturbation of the stationary state can be resolved and evolved well. In order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact scheme are used to construct different WCN schemes. In addition, the local characteristic projections are considered to further restrain the potential numerical oscillations. A variety of representative one- and two-dimensional examples are tested to demonstrate the good performance of the proposed schemes.

We introduce a third order adaptive mesh method to arbitrary high order Godunov approach. Our adaptive mesh method consists of two parts, i.e., mesh-redistribution algorithm and solution algorithm. The mesh-redistribution algorithm is derived based on variational approach, while a new solution algorithm is developed to preserve high order numerical accuracy well. The feature of proposed Adaptive ADER scheme includes that 1). all simulations in this paper are stable for large CFL number, 2). third order convergence of the numerical solutions is successfully observed with adaptive mesh method, and 3). high resolution and non-oscillatory numerical solutions are obtained successfully when there are shocks in the solution. A variety of numerical examples show the feature well.

The Buckley–Leverett partial differential equation has long been used to model two-phase flow in porous media. In recent years, the PDE has been modified to include a rate-dependent capillary pressure constitutive equation, known as dynamic capillary pressure. Previous traveling wave analysis of the modified Buckley–Leverett equation uncovered non-classical solutions involving undercompressive shocks. More recently, thermodynamically constrained averaging theory (TCAT) has generalized the capillary pressure equation by including additional dependence on fluid properties. In this paper, the model and traveling wave analysis are updated to incorporate TCAT capillary pressure as a generalization of dynamic capillary pressure. Solutions of the corresponding Riemann problem are similar to previous results except in the physically relevant situation in which both phases are pure fluids. The results presented here shed new light on the nature of the interface between one pure fluid displacing another pure fluid, in accordance with TCAT.

In this paper, a high-order curved mesh generation method for Discontinuous Galerkin methods is introduced. First, a regular mesh is generated. Second, the solid surface is re-constructed using cubic polynomial. Third, the elastic governing equations are solved using high-order finite element method to provide a fully or partly curved grid. Numerical tests indicate that the intersection between element boundaries can be avoided by carefully defining the elasticity modulus.

This article is devoted to analyze some ambiguities coming from a class of sediment transport models. The models under consideration are governed by the coupling between the shallow-water and the Exner equations. Since the PDE system turns out to be an hyperbolic system in non conservative form, ambiguities may occur as soon as the solution contains shock waves. To enforce a unique definition of the discontinuous solutions, we adopt the path-theory introduced by Dal Maso, LeFLoch and Murat [18]. According to the path choices, we exhibit several shock definitions and we prove that a shock with a constant propagation speed and a given left state may connect an arbitrary right state. As a consequence, additional assumptions (coming from physical considerations or other arguments) must be chosen to enforce a unique definition. Moreover, we show that numerical ambiguities may still exist even when a path is chosen to select the system's solution.

In this paper, we develop and study numerical methods for the two-mode shallow water equations recently proposed in [S. STECHMANN, A. MAJDA, and B. KHOUIDER, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407-432]. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches—two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme—and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method.

The formation of singularities in relativistic flows is not well understood. Smooth solutions to the relativistic Euler equations are known to have a finite lifespan; the possible breakdown mechanisms are shock formation, violation of the subluminal conditions and mass concentration. We propose a new hybrid Glimm/central-upwind scheme for relativistic flows. The scheme is used to numerically investigate, for a family of problems, which of the above mechanisms is involved.

The numerical simulation of non conservative system is a difficult challenge for two reasons at least. The first one is that it is not possible to derive jump relations directly from conservation principles, so that in general, if the model description is non ambiguous for smooth solutions, this is no longer the case for discontinuous solutions. From the numerical view point, this leads to the following situation: if a scheme is stable, its limit for mesh convergence will depend on its dissipative structure. This is well known since at least [1]. In this paper we are interested in the “dual” problem: given a system in non conservative form and consistent jump relations, how can we construct a numerical scheme that will, for mesh convergence, provide limit solutions that are the exact solution of the problem. In order to investigate this problem, we consider a multiphase flow model for which jump relations are known. Our scheme is an hybridation of Glimm scheme and Roe scheme.

In this paper, high-order Discontinuous Galerkin (DG) method is used to solve the two-dimensional Euler equations. A shock-capturing method based on the artificial viscosity technique is employed to handle physical discontinuities. Numerical tests show that the shocks can be captured within one element even on very coarse grids. The thickness of the shocks is dominated by the local mesh size and the local order of the basis functions. In order to obtain better shock resolution, a straightforward hp-adaptivity strategy is introduced, which is based on the high-order contribution calculated using hierarchical basis. Numerical results indicate that the hp-adaptivity method is easy to implement and better shock resolution can be obtained with smaller local mesh size and higher local order.

The space-time conservation element and solution element (CE/SE) method is proposed for solving a conservative interface-capturing reduced model of compressible two-fluid flows. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term for accounting the energy exchange. The one and two-dimensional flow models are numerically investigated in this manuscript. The CE/SE method is capable to accurately capture the sharp propagating wavefronts of the fluids without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building block of the suggested method. The method differs from the previous techniques because of global and local flux conservation in a space-time domain without resorting to interpolation or extrapolation. In order to reveal the efficiency and performance of the approach, several numerical test cases are presented. For validation, the results of the current method are compared with other finite volume schemes.

Existence and admissibility of δ-shock solutions is discussed for the non-convex strictly hyperbolic system of equations

The system is fully nonlinear, i.e. it is nonlinear with respect to both unknowns, and it does not admit the classical Lax-admissible solution for certain Riemann problems. By introducing complex-valued corrections in the framework of the weak asymptotic method, we show that a compressive δ-shock solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued, and the solutions can be seen to fit into the framework of weak singular solutions defined by Danilov and Shelkovich. Indeed, in this context, we can show that every 2 × 2 system of conservation laws admits δ-shock solutions.

In this work, the modified ghost fluid method is developed to deal with 2D compressible fluid interacting with elastic solid in an Euler-Lagrange coupled system. In applying the modified Ghost Fluid Method to treat the fluid-elastic solid coupling, the Navier equations for elastic solid are cast into a system similar to the Euler equations but in Lagrangian coordinates. Furthermore, to take into account the influence of material deformation and nonlinear wave interaction at the interface, an Euler-Lagrange Riemann problem is constructed and solved approximately along the normal direction of the interface to predict the interfacial status and then define the ghost fluid and ghost solid states. Numerical tests are presented to verify the resultant method.

In this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the book Supersonic Flow and Shock Waves by Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed.