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I will discuss a new methodology for adding localized, spacetime smooth, artificial viscosity to nonlinear systems of conservation laws which propagate shock waves, contact discontinuities, and rarefaction waves. To the system of conservation laws, we couple a linear scalar reactiondiffusion 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 spacetime to zero away from discontinuities. In 1D, this approach can be viewed as a carefully chosen spacetime 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 leveli.e., the level of the partial differential equations (PDE) so that a variety of higherorder numerical discretization scheme can be employed, and (2) Riemannbased solvers and characteristic decompositions are not needed near strong shocks.
I will show numerical simulations of this methodology on a number of classical shocktube problems, and describe how this methodology can be used in the multiD setting. This is joint work with Jon Reisner and Jonny Serensca.
