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ADAPTIVE SPECTRAL VISCOSITY FOR HYPERBOLIC CONSERVATION LAWS EITAN TADMOR AND KNUT WAAGAN
 

Summary: ADAPTIVE SPECTRAL VISCOSITY FOR HYPERBOLIC CONSERVATION LAWS
EITAN TADMOR AND KNUT WAAGAN
Abstract. Spectral approximations to nonlinear hyperbolic conservation laws require dissipative regulariza-
tion for stability. The dissipative mechanism must on the other hand be small enough, in order to retain the
spectral accuracy in regions where the solution is smooth. We introduce a new form of viscous regularization
which is activated only in the local neighborhood of shock discontinuities. The basic idea is to employ a
spectral edge detection algorithm as a dynamical indicator of where in physical space to apply numerical
viscosity. The resulting spatially local viscosity is successfully combined with spectral viscosity, where a much
higher than usual cut-off frequency can be used. Numerical results show that the new Adaptive Spectral
Viscosity scheme significantly improves the accuracy of the standard spectral viscosity scheme. In particu-
lar, results are improved near the shocks and at low resolutions. Examples include numerical simulations of
Burgers' equation, shallow water with bottom topography and the isothermal Euler equations. We also test
the schemes on a non-convex scalar problem, finding that the new scheme approximates the entropy solution
more reliably than the standard spectral viscosity scheme.
1. Introduction - main ingredients
Let u() : (-, ) R be a 2-periodic function and let uN PN u(x) denote its pseudospectral approxi-
mation of order N,
(1.1a) uN (x) = PN u(x) :=
N
k=-N

  

Source: Anisimov, Mikhail - Institute for Physical Science and Technology & Department of Chemical Engineering and Biomolecular Engineering, University of Maryland at College Park

 

Collections: Physics; Materials Science