Global Structure Stability of Riemann Solution of Quasilinear Hyperbolic System of Conservation Law in Presence of Boundary： Shocks and Contact Discontinuities

In this paper, we investigate a class of mixed initial-boundary value problems for a kind of n × n quasilinear hyperbolic systems of conservation laws on the quarter plan. We show that the structure of the pieeewise C^1 solution u = u（t, x） of the problem, which can be regarded as a perturbation of the corresponding Riemann problem, is globally similar to that of the solution u = U（x/t） of the corresponding Riemann problem. The piecewise C^1 solution u = u（t, x） to this kind of problems is globally structure-stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

二维交错网格的GAUSS型格式

利用Gauss型求积公式在交错网格的情况下构造了一类不需解Riemann问题的求解二维双曲守恒律的二阶显式Gauss型差分格式,该格式在CFL条件限制下为MmB格式.并将格式推广到二维方程组,进行了数值试验.     

A higher-order macroscopic model for pedestrian flows

This paper develops a higher-order macroscopic model of pedestrian crowd dynamics derived from fluid dynamics that consists of two-dimensional Euler equations with relaxation. The desired directional motion of pedestrians is determined by an Eikonal-type equation, which describes a problem that minimizes the instantaneous total walking cost from origin to destination. A linear stability analysis of the model demonstrates its ability to describe traffic instability in crowd flows. The algorithm to solve the macroscopic model is composed of a splitting technique introduced to treat the relaxation terms, a second-order positivity-preserving central-upwind scheme for hyperbolic conservation laws, and a fast-sweeping method for the Eikonal-type equation on unstructured meshes. To test the applicability of the model, we study a challenging pedestrian crowd flow problem of the presence of an obstruction in a two-dimensional continuous walking facility. The numerical results indicate the rationality of the model and the effectiveness of the computational algorithm in predicting the flux or density distribution and the macroscopic behavior of the pedestrian crowd flow. The simulation results are compared with those obtained by the two-dimensional Lighthill-Whitham-Richards pedestrian flow model with various model parameters, which further shows that the macroscopic model is able to correctly describe complex phenomena such as “stop-and-go waves” observed in empirical pedestrian flows.     

A numerical study of a rotationally degenerate hyperbolic system. Part I. The Riemann problem

We study numerically the Riemann problem for a 2 x 2 system of conservation laws with a cubic flux function, a particular case of the class of models introduced by Keyfitz and Kranzer. The system is not strictly hyperbolic, and the classical Lax theory for hyperbolic systems is not directly applicable. Correspondingly, some numerical schemes which are accurate for strictly hyperbolic systems are not well behaved for this example. When they do work, different schemes yield markedly different results for certain data. We explain this effect by observing that, near these data, viscous regularization is non-uniform as the viscosity tends to zero. This fact does not contradict the well-posedness of the hyperbolic model; it does imply that precise control of the viscosity introduced into a computational method is crucial for generating the correct numerical solutions. We examine all of these issues and comment on their implications for similar systems which arise in continuum mechanics.     

Density Large Deviations for Multidimensional Stochastic Hyperbolic Conservation Laws

We investigate the density large deviation function for a multidimensional conservation law in the vanishing viscosity limit, when the probability concentrates on weak solutions of a hyperbolic conservation law. When the mobility and diffusivity matrices are proportional, i.e. an Einstein-like relation is satisfied, the problem has been solved in Bellettini and Mariani (Bull Greek Math Soc 57:31–45, 2010). When this proportionality does not hold, we compute explicitly the large deviation function for a step-like density profile, and we show that the associated optimal current has a non trivial structure. We also derive a lower bound for the large deviation function, valid for a more general weak solution, and leave the general large deviation function upper bound as a conjecture.     

Global Structure Stability of Riemann Solution of Quasilinear Hyperbolic System of Conservation Law in Presence of Boundary: Shocks and Contact Discontinuities

In this paper, we investigate a class of mixed initial-boundary value problems for a kind of n×n quasilinear hyperbolic systems of conservation laws on the quarter plan. We show that the structure of the piecewise C¹ solution u=u(t,x) of the problem, which can be regarded as a perturbation of the corresponding Riemann problem, is globally similar to that of the solution u=U(x/t) of the corresponding Riemann problem. The piecewise C¹ solution u=u(t,x) to this kind of problems is globally structure-stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws

We study the large-time behaviors of solutions of viscous conservation laws whose inviscid part is a nonstrictly hyperbolic system. The initial data considered here is a perturbation of a constant state. It is shown that the solutions converge to single-mode diffusion waves in directions of strictly hyperbolic fields, and to multiple-mode diffusion waves in directions of nonstrictly hyperbolic fields. The multiple-mode diffusion waves, which are the new elements here, are the self-similar solutions of the viscous conservation laws projected to the nonstrictly hyperbolic fields, with the nonlinear fluxes replaced by their quadratic parts. The convergence rate to these diffusion waves isO(t ^–3/4+1/2p+) inL ^p , 1p, with >0 being arbitrarily small.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38     

Symmetry Analysis and Conservation Laws for the Hunter-Saxton Equation

In this paper,the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation(HSE) is analyzed.By applying the basic Lie symmetry method for the HSE,the classical Lie point symmetry operators are obtained.Also,the algebraic structure of the Lie algebra of symmetries is discussed and an optimal system of onedimensional subalgebras of the HSE symmetry algebra which creates the preliminary classification of group invariant solutions is constructed.Particularly,the Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained.Mainly,the conservation laws of the HSE are computed via three different methods including Boyer's generalization of Noether's theorem,first homotopy method and second homotopy method.     

A new view of the ECSK theory with a Dirac spinor

A formulation of the ECSK (Einstein-Cartan-Sciama-Kibble) theory with a Dirac spinor is given in terms of differential forms with values in exterior vector bundles associated with a fixed principalSL(2, )-bundle over a 4-manifold. In particular, tetrad fields are represented as soldering forms. In this setting, both the scalar curvature (Einstein-Hilbert) action density and the Dirac action density are well-defined polynomial functions of the soldering form and an independentSL(2,)-connection form. Thus, these densities are defined even where the tetrad field is degenerate (e.g. when fluctuations in the gravitational field are large). A careful analysis of the initial-value problem (in terms of an evolving triad field, SU(2)-connection, second-fundamental form and spinor field) reveals a first-order hyperbolic system of 27 evolution equations (not including the 8 evolution equations for the Dirac spinor) and 16 constraints. There are 10 conservation equations (due to local Poincaré invariance) which team up with some of the evolution equations to guarantee that the 16 constraints are preserved under the evolution.     

自适应间断有限元方法求解双曲守恒律方程

研究自适应Runge-Kutta间断Galerkin (RKDG)方法求解双曲守恒律方程组,并提出两种生成相容三角形网格的自适应算法.第一种算法适用于规则网格,实现简单、计算速度快.第二种算法基于非结构网格,设计一类基于间断界面的自适应网格加密策略,方法灵活高效.两种方法都具有令人满意的计算效果,而且降低了RKDG的计算量.     

Compact Accurately Boundary-Adjusting high-REsolution Technique for fluid dynamics

A novel high-resolution numerical method is presented for one-dimensional hyperbolic problems based on the extension of the original Upwind Leapfrog scheme to quasi-linear conservation laws. The method is second-order accurate on non-uniform grids in space and time, has a very small dispersion error and computational stencil defined within one space–time cell. For shock-capturing, the scheme is equipped with a conservative non-linear correction procedure which is directly based on the maximum principle. Plentiful numerical examples are provided for linear advection, quasi-linear scalar hyperbolic conservation laws and gas dynamics and comparisons with other computational methods in the literature are discussed.     

13-Moment System with Global Hyperbolicity for Quantum Gas

We point out that the quantum Grad’s 13-moment system (Yano in Physica A 416:231–241, 2014) is lack of global hyperbolicity, and even worse, the thermodynamic equilibrium is not an interior point of the hyperbolicity region of the system. To remedy this problem, by fully considering Grad’s expansion, we split the expansion into the equilibrium part and the non-equilibrium part, and propose a regularization for the system with the help of the new hyperbolic regularization theory developed in Cai et al. (SIAM J Appl Math 75(5):2001–2023, 2015) and Fan et al. (J Stat Phys 162(2):457–486, 2016). This provides us a new model which is hyperbolic for all admissible thermodynamic states, and meanwhile preserves the approximate accuracy of the original system. It should be noted that this procedure is not a trivial application of the hyperbolic regularization theory.     

A Second-Order Three-Level Difference Scheme for a Magneto-Thermo-Elasticity Model

This article deals with the numerical solution to the magneto-thermo-elasticity model, which is a system of the third order partial differential equations. By introducing a new function, the model is transformed into a system of the second order generalized hyperbolic equations. A priori estimate with the conservation for the problem is established. Then a three-level finite difference scheme is derived. The unique solvability, unconditional stability and second-order convergence in $L_{\infty}$-norm of the difference scheme are proved. One numerical example is presented to demonstrate the accuracy and efficiency of the proposed method.     

A Cartesian grid embedded boundary method for hyperbolic conservation laws

We present a second-order Godunov algorithm to solve time-dependent hyperbolic systems of conservation laws on irregular domains. Our approach is based on a formally consistent discretization of the conservation laws on a finite-volume grid obtained from intersecting the domain with a Cartesian grid. We address the small-cell stability problem associated with such methods by hybridizing our conservative discretization with a stable, nonconservative discretization at irregular control volumes, and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation. The resulting method is second-order accurate in L¹ for smooth problems, and is robust in the presence of large-amplitude discontinuities intersecting the irregular boundary.     

Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere’s tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here “equatorial periodic solutions”, analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct “confined solutions”, which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test cases are presented.     

Entropy of averaging for compressible two-pressure two-phase flow models

We propose here a new closure for compressible two-pressure two-phase flow models, which satisfies conservation requirements, boundary conditions at the edges of the mixing zone, hyperbolic stability (real eigenvalues for the characteristic version of the equations of motion) and an entropy inequality. Except for the latter, these properties are direct consequences of the proposed closures. The entropy, which is the main focus of this Letter, inequality (as opposed to entropy conservation for microphysically adiabatic processes) implies positivity for the entropy of averaging.     

Quasilinear hyperbolic systems

We construct global solutions for quasilinear hyperbolic systems and study their asymptotic behaviors. The systems include models of gas flows in a variable area duct and flows with a moving source. Our analysis is based on a numerical scheme which generalizes the Glimm scheme for hyperbolic conservation laws.Partially supported by National Science Foundation Grant NSF MCS78-2202     

Phase diagrams in unidirectionally coupled map lattice for open traffic flow

The jamming transition from the free traffic to the oscillatory traffic is investigated with the unidirectionally coupled map lattice model which has the hyperbolic tangent local map. Spatio-temporal structures in the jamming transition are found with the use of numerical simulation. The traffic states are studied for both constant and noisy boundary conditions. We show the phase diagrams of different kinds of congested traffic. It is found that the noise at the boundary has an important effect on the traffic states. The traffic behavior in the coupled map lattice model exhibits a jamming transition similar to that found in the car-following model.     

A Diffusively Corrected Multiclass Lighthill-Whitham-Richards Traffic Model with Anticipation Lengths and Reaction Times

Multiclass Lighthill-Whitham-Richards traffic models [Benzoni-Gavage and Colombo, Euro. J. Appl. Math., 14 (2003), pp. 587–612; Wong and Wong, Transp. Res. A, 36 (2002), pp. 827–841] give rise to first-order systems of conservation laws that are hyperbolic under usual conditions, so that their associated Cauchy problems are well-posed. Anticipation lengths and reaction times can be incorporated into these models by adding certain conservative second-order terms to these first-order conservation laws. These terms can be diffusive under certain circumstances, thus, in principle, ensuring the stability of the solutions. The purpose of this paper is to analyze the stability of these diffusively corrected models under varying reaction times and anticipation lengths. It is demonstrated that instabilities may develop for high reaction times and short anticipation lengths, and that these instabilities may have controlled frequencies and amplitudes due to their nonlinear nature.     

Accelerating cosmologies from compactification

A solution of the (4+n)-dimensional vacuum Einstein equations is found for which spacetime is compactified on an n-dimensional compact hyperbolic manifold (n> or =2) of time-varying volume to a flat four-dimensional Friedmann-Lemaitre-Robertson-Walker cosmology undergoing a period of accelerated expansion in the Einstein conformal frame. This shows that the "no-go" theorem forbidding acceleration in "standard" (time-independent) compactifications of string or M theory does not apply to "cosmological" (time-dependent) hyperbolic compactifications.