The hierarchical reconstruction (HR) [Liu, Shu, Tadmor and Zhang, SINUM ’07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservati...

Hyperbolic conservation laws posed on manifolds arise in many applications to geophysical flows andgeneral relativity. Recentworkby the author and his collaborators attempts to set the foundations for a study of weak solutions defined on Riemannian or Lorentzian manifolds and includes an investigation of the existence and qualitative behavior of solutions. The metric on the manifold may either ...

In this paper the δ-shock front problem is studied. For some classes of hyperbolic systems of conservation laws (in several space dimension, too) we introduce the definitions of a δ-shock wave type solution relevant to the front problem. The Rankine–Hugoniot conditions for δ-shocks are analyzed from both geometrical and physical points of view. δ-Shock balance relations connected with area and ...

In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of Hamilton-Jacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since Hamilton-Jacobi equations in general...

In {J. Comput. Phys. 229 (2010) 8105-8129}, we studied hybrid weighted essentially non-oscillatory (WENO) schemes with different indicators for hyperbolic conservation laws on uniform grids for Cartesian domains. In this paper, we extend the schemes to solve two-dimensional systems of hyperbolic conservation laws on curvilinear grids for non-Cartesian domains. Our goal is to obtain similar adva...

This paper consider various examples of metrics which are contractive w.r.t. an evolution semigroup, and discusses the possibility of an abstract O.D.E. theory on metric spaces, with applications to hyperbolic systems. In particular, using a recently introduced deenition of Viscosity Solutions, it is shown how a strictly hyperbolic system of conservation laws can be reformulated as an abstract ...

We explicitly construct a convex entropy function for a hyperbolic system with relaxation. This entropy is deened in the sense of Chen, Levermore and Liu 2], which resembles Boltzman's H-Theorem for kinetic equations. This construction follows the idea of Suliciu's energy function for a rate-type mixed hyperbolic-elliptic system 10]. Such an entropy will be useful in proving the entropy propert...

We consider two new classes of examples of sup-norm blowup in finite time for strictly hyperbolic systems of conservation laws. The explosive growth in amplitude is caused either by a gradient catastrophe or by a singularity in the flux function. The examples show that solutions of (uniformly) strictly hyperbolic systems can remain as smooth as the initial data until the time of blowup. Consequ...

A necessary and sufficient condition is given for all hyperbolic difference schemes that use up to nine mesh points to be of second-order accuracy. We also construct a new difference scheme for two-dimensional hyperbolic systems of conservation laws. The scheme is of second-order accuracy and requires knowledge of only seven mesh points. A stability condition is obtained and is utilized in nume...

The system of shallow water waves is one of the classical examples for nonlinear, twodimensional conservation laws. The paper investigates a simple kinetic equation depending on a parameter " which leads for " ! 0 to the system of shallow water waves. The corresponding 'equilibrium' distribution function has a compact support which depends on the eigenvalues of the hyperbolic system. It is show...