This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a con...This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under the condition that the flux function has a finite number of weak discontinuous points, by using the structure of weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary, we clarify the geometric structure and the behavior of boundary for the weak entropy solution.展开更多
In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and the numerical simu...In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and the numerical simulations were given.展开更多
This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um ar...This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um are three given constants satisfying um=u+≠u- or um=u-≠u+ , and the flux function f is a given continuous function with a weak discontinuous point ud. The main purpose of our present manuscript is devoted to studying the structure of the global weak entropy solution for the above initial-boundary value problem under the condition of f '-(ud) > f '+(ud). By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial-boundary value problem, and investigate the interaction of elementary waves with the boundary and the boundary behavior of the weak entropy solution.展开更多
This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws.The basic idea is that the“meaningful objects”are the fluxes,evaluate...This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws.The basic idea is that the“meaningful objects”are the fluxes,evaluated across domain boundaries over time intervals.The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting.It implies that a weak solution indeed satisfies the balance law.In fact,it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary.It should be emphasized that the weak solutions considered here need not be entropy solutions.Furthermore,the assumption imposed on the flux f(u)is quite minimal-just that it is locally bounded.展开更多
Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux ...Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measurevalued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.展开更多
In this paper, we first summarize several applications of the flux approximation method on hyperbolic conservation systems. Then, we introduce two hyperbolic conservation systems (2.1) and (2.2) of Temple’s type, and...In this paper, we first summarize several applications of the flux approximation method on hyperbolic conservation systems. Then, we introduce two hyperbolic conservation systems (2.1) and (2.2) of Temple’s type, and prove that the global weak solutions of each system could be obtained by the limit of the linear combination of two systems.展开更多
The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t + f(k(x, t), u)_x = 0,where the coefficient k(x, t...The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t + f(k(x, t), u)_x = 0,where the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t)plane. In contrast to most of the existing literature on problems with discontinuouscoefficients, here the convergence proof is not based on the singular mapping approach,but rather on the div-curl lemma (but not the Young measure) and a Lax type en-tropy estimate that is robust with respect to the regularity of k(x, t). Following [14],the authors propose a definition of entropy solution that extends the classical Kruzkovdefinition to the situation where k(x, t) is piecewise Lipschitz continuous in the (x, t)plane, and prove the stability (uniqueness) of such entropy solutions, provided that theflux function satisfies a so-called crossng condition, and that strong traces of the solu-tion exist along the curves where k(x, t) is discontinuous. It is shown that a convergentsubsequence of approximations produced by the Lax-Friedrichs scheme converges tosuch an entropy solution, implying that the entire computed sequence converges.展开更多
This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. B...This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.展开更多
In this paper,we describe a numerical technique for the solution of macroscopic traffic flow models on networks of roads.On individual roads,we consider the standard Lighthill-Whitham-Richards model which is discretiz...In this paper,we describe a numerical technique for the solution of macroscopic traffic flow models on networks of roads.On individual roads,we consider the standard Lighthill-Whitham-Richards model which is discretized using the discontinuous Galerkin method along with suitable limiters.To solve traffic flows on networks,we construct suitable numerical fluxes at junctions based on preferences of the drivers.We prove basic properties of the constructed numerical flux and the resulting scheme and present numerical experiments,including a junction with complicated traffic light patterns with multiple phases.Differences with the approach to numerical fluxes at junctions fromČanićet al.(J Sci Comput 63:233-255,2015)are discussed and demonstrated numerically on a simple network.展开更多
文摘This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under the condition that the flux function has a finite number of weak discontinuous points, by using the structure of weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary, we clarify the geometric structure and the behavior of boundary for the weak entropy solution.
基金Project supported by National Natural Science Foundation of China(Grant No .10271072)
文摘In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and the numerical simulations were given.
文摘This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um are three given constants satisfying um=u+≠u- or um=u-≠u+ , and the flux function f is a given continuous function with a weak discontinuous point ud. The main purpose of our present manuscript is devoted to studying the structure of the global weak entropy solution for the above initial-boundary value problem under the condition of f '-(ud) > f '+(ud). By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial-boundary value problem, and investigate the interaction of elementary waves with the boundary and the boundary behavior of the weak entropy solution.
基金the Institute of Applied Physics and Computational Mathematics,Beijing,for the hospitality and support.The second author is supported by the NSFC(Nos.11771054,12072042,91852207)the Sino-German Research Group Project(No.GZ1465)the National Key Project GJXM92579.
文摘This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws.The basic idea is that the“meaningful objects”are the fluxes,evaluated across domain boundaries over time intervals.The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting.It implies that a weak solution indeed satisfies the balance law.In fact,it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary.It should be emphasized that the weak solutions considered here need not be entropy solutions.Furthermore,the assumption imposed on the flux f(u)is quite minimal-just that it is locally bounded.
基金supported by the Research Council of Norway through theprojects Nonlinear Problems in Mathematical Analysis Waves In Fluids and Solids+2 种基金 Outstanding Young Inves-tigators Award (KHK), the Russian Foundation for Basic Research (grant No. 09-01-00490-a) DFGproject No. 436 RUS 113/895/0-1 (EYuP)
文摘Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measurevalued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.
文摘In this paper, we first summarize several applications of the flux approximation method on hyperbolic conservation systems. Then, we introduce two hyperbolic conservation systems (2.1) and (2.2) of Temple’s type, and prove that the global weak solutions of each system could be obtained by the limit of the linear combination of two systems.
基金Project supported by the BeMatA Program of the Research Council of Norway and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282
文摘The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t + f(k(x, t), u)_x = 0,where the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t)plane. In contrast to most of the existing literature on problems with discontinuouscoefficients, here the convergence proof is not based on the singular mapping approach,but rather on the div-curl lemma (but not the Young measure) and a Lax type en-tropy estimate that is robust with respect to the regularity of k(x, t). Following [14],the authors propose a definition of entropy solution that extends the classical Kruzkovdefinition to the situation where k(x, t) is piecewise Lipschitz continuous in the (x, t)plane, and prove the stability (uniqueness) of such entropy solutions, provided that theflux function satisfies a so-called crossng condition, and that strong traces of the solu-tion exist along the curves where k(x, t) is discontinuous. It is shown that a convergentsubsequence of approximations produced by the Lax-Friedrichs scheme converges tosuch an entropy solution, implying that the entire computed sequence converges.
基金the NSF-Guangdong China(04010473)Jinan University Foundation(51204033)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry(No.2005-383)
文摘This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.
基金The work of L.Vacek is supported by the Charles University,project GA UK No.1114119The work of V.Kučera is supported by the Czech Science Foundation,project No.20-01074S.
文摘In this paper,we describe a numerical technique for the solution of macroscopic traffic flow models on networks of roads.On individual roads,we consider the standard Lighthill-Whitham-Richards model which is discretized using the discontinuous Galerkin method along with suitable limiters.To solve traffic flows on networks,we construct suitable numerical fluxes at junctions based on preferences of the drivers.We prove basic properties of the constructed numerical flux and the resulting scheme and present numerical experiments,including a junction with complicated traffic light patterns with multiple phases.Differences with the approach to numerical fluxes at junctions fromČanićet al.(J Sci Comput 63:233-255,2015)are discussed and demonstrated numerically on a simple network.
