On exact dimensional splitting for a multidimensional scalar quasilinear hyperbolic conservation law

A dimensional splitting scheme is applied to a multidimensional scalar homogeneous quasilinear hyperbolic equation (conservation law). It is proved that the splitting error is zero. The proof is presented for the above partial differential equation in an arbitrary number of dimensions. A numerical example is given that illustrates the proved accuracy of the splitting scheme. In the example, the grid convergence of split (locally one-dimensional) compact and bicompact difference schemes and unsplit bicompact schemes combined with high-order accurate time-stepping schemes (namely, Runge–Kutta methods of order 3, 4, and 5) is analyzed. The errors of the numerical solutions produced by these schemes are compared. It is shown that the orders of convergence of the split schemes remain high, which agrees with the conclusion that the splitting error is zero.     

A solution formula for a non-convex scalar hyperbolic conservation law with monotone initial data

In this paper we prove an explicit representation formula for the solution of a one-dimensional hyperbolic conservation law with a non-convex flux function but monotone initial data. This representation formula is similar to those of Lax [10] and Kunik [7,8] and enables us to compute the solution pointwise explicitly. This result is a generalization of a theorem given in Kunik [8] where the case of only one inflexion point for the fluxes was considered. Its proof uses the polygonal method of Dafermos [2]. The application of this method leads to a simple explicit construction of the solutions for a Kynch sedimentation process [9] and to an explicit parameter representation for the shock curves evolving during the sedimentation process.     

On the optimization of a conservation law

Consider the Cauchy problem for a conservation law and assume that an integral functional on its solution is defined. In this note we obtain an Euler-Lagrange equation for the stationary points of this functional. An application to the optimal management of traffic flows is considered.Received: 29 April 2002, Accepted: 24 October 2002, Published online: 4 September 2003Mathematics Subject Classification (2000): 35L65, 49K20, 90B20The authors thank Stefano Bianchini for useful discussions.     

A numerical method for some initial-value problems of one scalar hyperbolic conservation law

We examine the existence and regularity results for a scalar conservation law with a convexity condition and solve its weak solution with shocks by using a special method of characterization combined with a representation formula for the weak solution.     

On the mixed problem for hyperbolic regularizations of conservation laws

The mixed problem for hyperbolic regularization of conservation laws is studied.     

关于一个非线性几何光学中的非严格双曲守恒律系统的研究

研究了一个产生于非线性几何光学中的非严格双曲守恒律系统.该系统具有强非线性流函数项,且狄拉克激波可能同时出现在解的两个状态变量中.通过未知函数的一个变换,该系统的非线性流函数项得到弱化,从而其黎曼问题被完全解决.     

Regularity for a singular conservation law

The main structure underlying the nonlinearity of conservation laws of gasdynamical type in two independent variables will be discussed at the hand of a canonical example describing also properties of water waves near shore. The ultimately singular nature of such laws is here the central issue and calls for an unusual formulation (Sect. 2). Attention is directed to the globally strong solutions, and an unusual regularization (Sect. 2) is employed to make them accessible, after illposedness is overcome (Sect. 3). The usual regularity theory is not normally sufficient for singular partial differential equations, and the necessary additional chapter on extensions to the singular locus is developed (Sect. 4) in detail for the canonical example. Criteria for the relation between regularized and strong solutions are discussed in Section 5 and used in Section 6 to characterize the class of solutions that are globally strong in the strictest sense.     

On the optimization of flux limiter schemes for hyperbolic conservation laws

A classic strategy to obtain high‐quality discretizations of hyperbolic partial differential equations is to use flux limiter (FL) functions for blending two types of approximations: a monotone first‐order scheme that deals with discontinuous solution features and a higher order method for approximating smooth solution parts. In this article, we study a new approach to FL methods. Relying on a classification of input data with respect to smoothness, we associate specific basis functions with the individual smoothness notions. Then, we construct a limiter as a linear combination of the members of parameter‐dependent families of basis functions, and we explore the possibility to optimize the parameters in interesting model situations to find a corresponding optimal limiter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On the riemann problem for a prototype of a mixed type conservation law

We study the system of conservation laws given by With initial value The system is elliptic when u² + v² < ρ² and hyperbolic when u² + v² ≧ ρ². Following Liu's construction it is found that the system always has a weak solution which however is not necessarily unique.     

On the convergence of a local third order shock capturing method for hyperbolic conservation laws

The Piecewise Polynomial Harmonic Method (PPHM) is a local third order accurate shock capturing method for hyperbolic conservation laws. In this paper, theoretical stability properties are presented. Using these properties, the convergence of the scheme for scalar conservation laws is obtained. A direct adaptation of these results should be used to derive the convergence of the PHM (Marquina, SIAM J. Sci. Comput. 15(4), 892–915, 1994). Finally, the method is tested in several classical problems in order to explore its numerical behavior.     

On the optimization of the initial boundary value problem for a conservation law

In the present note, the theory of shift differentiability for the Cauchy problem is extended to the case of an initial boundary value problem for a conservation law. This result allows to exhibit an Euler-Lagrange equation to be satisfied by the extrema of integral functionals defined on the solutions of initial boundary value problems of this kind.     

Artifical and physical viscosity solutions for a hyperbolic conservation system

In this paper, a new maximum principle is introduced to study positive lower bounds of the density both for the artificial viscosity solutions and for the physical viscosity solutions of a 'hyperbolic conservationsystem derived from the Broadwell model and the global existence of theseviscosity solutions is obtained. We give some simple numerical results fordiscontinuous initial data.     

Asymptotic states for hyperbolic conservation laws with a moving source

We construct noninteracting wave patterns (i.e., asymptotic states) for a conservation law with a general moving source term. When nonlinear resonance occurs, which is the case when the characteristic speed is near the speed of the source, instability may result. We identify a stability criterion which is independent of the flux function. This is so, even if composite wave patterns exist, as may be the case for nonconvex flux functions. We study the general scalar model as well as transonic gas flows through a duct with varying cross section. For the latter case, noninteracting wave patterns for such a flow are constructed for arbitrary equations of state. It is shown that the stability of a wave pattern depends on the geometry of the duct, and not on the equation of the state. In particular, transonic steady shock waves along a converging duct are unstable, and flow along a diverging duct is always stable.     

On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms

We deal in this study with the convergence of a class of numerical schemes for scalar conservation laws including stiff source terms. We suppose that the source term is dissipative but it is not necessarily a Lipschitzian function. The convergence of the approximate solution towards the entropy solution is established for first and second order accurate MUSCL and for splitting semi-implicit methods.

     

Convergence of a relaxation scheme for hyperbolic systems of conservation laws

Summary. This paper concerns the study of a relaxation scheme for hyperbolic systems of conservation laws. In particular, with the compensated compactness techniques, we prove a rigorous result of convergence of the approximate solutions toward an entropy solution of the equilibrium system, as the relaxation time and the mesh size tend to zero. Received September 29, 1998 / Revised version received December 20, 1999 / Published online August 24, 2000     

On decay of periodic entropy solutions to a scalar conservation law

We establish a necessary and sufficient condition for decay of periodic entropy solutions to a multidimensional conservation law with merely continuous flux vector.     

On the reservoir technique convergence for nonlinear hyperbolic conservation laws - Part I

This paper is devoted to the analysis of flux schemes coupled with the reservoir technique for approximating hyperbolic equations and linear hyperbolic systems of conservation laws [F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, The reservoir scheme for systems of conservation laws, in: Finite Volumes for Complex Applications, III, Porquerolles, 2002, Lab. Anal. Topol. Probab. CNRS, Marseille, 2002, pp. 247-254 (electronic); F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, Un procédé de réduction de la diffusion numérique des schémas à différence de flux d'ordre un pour les systèmes hyperboliques non linéaires, C. R. Math. Acad. Sci. Paris 335 (7) (2002) 627-632; F. Alouges, F. De Vuyst, G. Le Coq, E. Lorin, The reservoir technique: A way to make Godunov-type schemes zero or very low diffusive. Application to Colella-Glaz, Eur. J. Mech. B Fluids 27 (6) (2008)]. We prove the long time convergence of the reservoir technique and its TVD property for some specific but still general configurations. Proofs are based on a precise study of the treatment by the reservoir technique of shock and rarefaction waves.