Computational and Applied Math Proseminar

Department of Mathematics, Arizona State University

Thurday, November 30, 2000, 12:15 p.m. in GWC Room 604

Eitan Tadmor

Department of Mathematics UCLA

High Resolution Central Schemes

Abstract We discuss recent developments of high-resolution schemes for approximate solution of hyperbolic conservation laws, Hamilton-Jacobi (HJ) e quations and related nonlinear problems. We focus on non-oscillatory central schemes as prototype for Godunov-type projection methods. A variety of numerical experiments demonstrate that the p roposed central schemes offer simple, robust, Riemann-solver-free ``black-box'' solvers, while at the same time, they retain the high-resolution content of the more expensive upwind schemes. Among other issues to be complemented by other presentations in this conference, we shall highlight the following topics:

$\bullet$ Scalar equations. Variation and entropy stability estimates as well as multidimensional $L^\infty$-bounds are presented.

$\bullet$ Systems of equations. Extension to systems is carried out by componentwise application of the scalar framework. It is in this context that the central schemes offer a remarkable advantage over the corresponding upwind framework.

$\bullet$ Multidimensional problems. Since we bypass the need for (approximate) Riemann solvers, multidimensional problems are solved without dimensional splitting. In fact, the proposed class of central schemes is utilized for a variety of nonlinear transport equations, and in this context we demonstrate the construction and implementation of central schemes for HJ and incompressible Euler equations.

$\bullet$ Semi-discrete formulations. We describe recent developments of new high-resolution central schemes for general convection-diffusion equations based on semi-discrete formulation of central schemes.

$\bullet$ Examples. We overview recent applications to various models, i ncluding incompressible flows, MHD equations, simulations of semi-conductors models, and more.

For further information please contact: mittelmann@asu.edu