Central Schemes: a Powerful Black-Box-Solver for Nonlinear Hyperbolic PDEs

Start: 10/23/2013 - 1:15pm
End  : 10/23/2013 - 2:15pm

Applied Math Seminar

Alexander Kurganov (Tulane University)


I will first give a brief description of finite-volume, Godunov-type methods for hyperbolic
systems of conservation laws. These methods consist of two types of schemes: upwind and
central. My lecture will focus on the second type -- non-oscillatory central schemes.

Godunov-type schemes are projection-evolution methods. In these methods, the solution, at
each time step, is interpolated by a (discontinuous) piecewise polynomial interpolant, which
is then evolved to the next time level using the integral form of conservation laws. Therefore,
in order to design an upwind scheme, (generalized) Riemann problems have to be
(approximately) solved at each cell interface. This however may be hard or even impossible.

The main idea in the derivation of central schemes is to avoid solving Riemann problems by
averaging over the wave fans generated at cell interfaces. This strategy leads to a family of
universal numerical methods that can be applied as a black-box-solver to a wide variety of
hyperbolic PDEs and related problems. At the same time, central schemes suffer from
(relatively) high numerical viscosity, which can be reduced by incorporating of some
upwinding information into the scheme derivation -- this leads to central-upwind schemes,
which will be presented in the lecture.

During the talk, I will show a number of recent applications of the central schemes.

Davidson, CMC