This paper describes a parallel adaptive strategy based on discontinuous hp-finite element approximations oflinear, scalar, hyperbolic conservation laws. The paper focuses on the development of an effective parallel adaptive strategy for such problems, Numerical experiments suggest that these techniques arc highly parallelizablc a.nd deliver super-linear rates of convergence, thereby yielding e...

We discuss the existence and uniqueness of discontinuous solutions to adjoint problems associated with nonlinear hyperbolic systems of conservation laws. By generalizing the Haar method for Glimm-type approximations to hyperbolic systems, we establish that entropy solutions depend continuously upon their initial data in the natural L norm.

We study a Riemann problem for the two-dimensional isentropic gas dynamics equations which models transonic regular reflection. When written in self-similar coordinates, the system changes type from hyperbolic to mixed hyperbolic-elliptic. Using the theory of one-dimensional hyperbolic conservation laws, we formulate a free boundary problem in the subsonic region and we outline the main ideas f...

Conservation laws are hyperbolic partial differential equations that are commonly used to express the fundamental balance laws that occur in many physical systems and engineering problems when dissipation effects are neglected. In these lectures we shall focus on systems that are made of the interconnection of scalar or 2x2 conservation laws in one space dimension. Typical examples include hydr...

Mathematical modelling of dynamical systems often yields partial differential equations (PDEs) in time and space, which represent a conservation law possibly including a source term. Uncertainties in physical parameters can be described by random variables. To resolve the stochastic model, the Galerkin technique of the generalised polynomial chaos results in a larger coupled system of PDEs. We ...

This is an expository paper discussing the regularity and large time behavior of admissible BV solutions of genuinely nonlinear, strictly hyperbolic systems of two conservation laws. The approach will be via the theory of generalized characteristics.

A modi ed triangle based adaptive di erence stencil for the numerical approximation of scalar hyperbolic conservation laws in two space dimensions is constructed. The scheme satis es the maximum principle and approximates the ux with second order accuracy. 3

The theory of hyperbolic conservation laws has been successfully applied to the study of vehicular traffic flows. We present here some models showing phase transitions, that in terms of traffic flows correspond to two distinct behaviors, free or congested.

We show that the two-component system of hyperbolic conservation laws ∂tρ + ∂x(ρu) = 0 = ∂tu + ∂xρ appears naturally in the formally computed hydrodynamic limit of some randomly growing interface models, and we study some properties of this system.

We analize the structure of the general solution of the Riemann problem for a striclty hyperbolic system of conservation laws whose characteristic elds are neither genuinely non-linear nor linearly degenerate in the sense of Lax.