where u : R × R → R is a vector of conserved variables (or state variables). For fluid dynamics, u is the vector of mass, momentum and energy denisties so that ∫ b a uj(x, t) dx is the total quantity of the j state variable in the interval at time t. Because these variables are conserved, ∫∞ −∞ uj(x, t) dx should be constant in t. The function f : R m → R is the flux function, which gives the r...

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 a conservation equation and definition of weak solutions. 2. Hyperbolic systems. Explicit solutions in the linear, constant coefficients case. Nonlinear effects: loss of regularity and wave interactions. 3. Shock wave...

This is a survey of interactions of weak nonlinear waves in N × N systems of hyperbolic conservation laws. Recently a variety of surprising new phenomena have been observed, including strong nonlinear instability of solutions. This implies that further assumptions must be made to develop a Glimm–Lax existence and decay theory for N ≥ 3. As a first step towards such a theory, a systematic descri...

As a new attempt to solve hyperbolic conservation laws with spatially varying fluxes, the weighted essentially non-oscillatory (WENO) method is applied to solve a multi-class traffic flow model for an inhomogeneous highway. The numerical scheme is based upon a modified equivalent system that is written in a “standard” hyperbolic conservation form. Numerical examples, which include the difficult...

We report on numerical experiments using adaptive sparse grid dis-cretization techniques for the numerical solution of scalar hyperbolic conservation laws. Sparse grids are an eecient approximation method for functions. Compared to regular, uniform grids of a mesh parameter h contain h ?d points in d dimensions, sparse grids require only h ?1 jloghj d?1 points due to a truncated , tensor-produc...

Spectral approximations to nonlinear hyperbolic conservation laws require dissipative regularization for stability. The dissipative mechanism must on the other hand be small enough, in order to retain the spectral accuracy in regions where the solution is smooth. We introduce a new form of viscous regularization which is activated only in the local neighborhood of shock discontinuities. The bas...

It is well known that the classic Galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the Euler equations for compressible flow. It is also well known that naively adding artificial diffusion to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. An elegant approach, referred to as spectral viscosity method...

We consider one-dimensional, locally finite interacting particle systems with two conservation laws. The models have a family of stationary measures with product structure and we assume the existence of a uniform bound on the inverse of the spectral gap which is quadratic in the size of the system. Under Eulerian scaling the hydrodynamic limit for the macroscopic density profiles leads to a two...

In this paper we consider the solution of hyperbolic conservation laws on moving meshes by means of an Arbitrary Lagrangian Eulerian (ALE) formulation. In particular we propose an ALE framework for the genuinely explicit residual distribution schemes of (Ricchiuto and Abgrall J.Comput.Phys 229, 2010). The discretizations obtained are thoroughly tested on a large number of benchmarks Key-words: ...

It is well known that the classic Galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the Euler equations for compressible flows. Adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use. An elegant solution developed in the context of spectral methods by Eitan Tadmor and coworkers is t...