The alternating evolution (AE) system of Liu [25] ∂tu +∇x · f(v) = 1 2 (v − u), ∂tv +∇x · f(u) = 1 2 (u− v) serves as a refined description of systems of hyperbolic conservation laws ∂tφ +∇x · f(φ) = 0, φ(x, 0) = φ0(x). The solution of conservation laws is precisely captured when two components take the same initial value as φ0, or approached by two components exponentially fast when 2 ↓ 0 if t...

We present a class of second-order conservative finite difference algorithms for solving numerically time-dependent problems for hyperbolic conservation laws in several space variables. These methods are upwind and multidimensional, in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge, and that all fluxes are evalua...

We consider the macroscopic model for traffic flow proposed by Aw and Rascle in 2000. The model is a 2×2 system of hyperbolic conservation laws, or, when the model includes a relaxation term, a 2 × 2 system of hyperbolic balance laws. The main difficulty is the presence of vacuum, which makes us unable to control the total variation of the conservative variables. We allow vacuum to appear and p...

This paper constructs multirate time discretizations for hyperbolic conservation laws that allow different time-steps to be used in different parts of the spatial domain. The discretization is second order accurate in time and preserves the conservation and stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global time-steps (restricted by...

These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: 1. Meaning of the conservation equations and definition of weak solutions. 2. Shocks, Rankine-Hugoniot equations and admissibility conditions. 3. Genuinely nonlinear and linearly degenerate characteristic fields. Solution to the Ri...

Ut + F (U)x = 0 correspond to the constitutive relations of an elastic medium if the flux F is determined by the value of the conserved quantity U(x, t). However, if viscosity or relaxation phenomena are present, this model is inadequate. In such a case, the flux depends also on the past history of the medium U(x, τ) for τ < t, i.e., the material has memory. An important class of media of this ...

A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robust...

1. INTRODUCI7ON We study certain Sobolev-type regularizations of the hyperbolic conservation laws that add terms simulating both dissipative and dispersive processes. These evolution equations have the form with the auxiliary specification The behaviour in L'(R) of the solutions of problem (S), as well as the value of (S) as an approximation to problem (C), is studied. Convergence results, with...

We analyze a class of L∞ vector fields, called divergence-measure fields. We establish the Gauss-Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of L∞ fields. Then we apply this theory to analyze L∞ entropy solutions of initial-boundary-value problems for hyperbolic conservation laws and to study the ways in which the solutions assume t...

2.1 Examples of conservative schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 The Godunov Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 The Lax-Friedrichs Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 The local Lax-Friedrichs Scheme . . . . . . . ....