 
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 cutoff 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 nonconvex 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 2periodic 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
