To the system of conservation laws, we couple a linear scalar reaction-diffusion equation whose solution C is used as an artificial modification of the flux vector, which determines both the location, localization, and strength of the artificial viscosity. Near shock discontinuities, C is large and localized, but transitions smoothlyin space-time to zero away from discontinuities. In 1-D, this approach can be viewed as a carefully chosen space-time smoothing of the classical artificial viscosity method devised by VonNeumann and Richtmyer (1950), and is provably convergent in the limit of zero mesh size.

This relatively simple approach has two fundamental features: (1) the regularization is at the continuum level--i.e., the level of the partial differential equations (PDE)-- so that a variety of higher-order numerical discretization scheme can be employed, and (2) Riemann-based solvers and characteristic decompositions are not needed near strong shocks.

I will show numerical simulations of this methodology on a number of classical shock-tube problems, and describe how this methodology can be used in the multi-D setting. This is joint work with Jon Reisner and Jonny Serensca.

Host: Misha Shashkov