Analysis of the spectral vanishing viscosity method for periodic conservation laws
SIAM Journal on Numerical Analysis
Convergence of spectral methods for nonlinear conservation laws
SIAM Journal on Numerical Analysis
The rate of convergence of spectral-viscosity methods for periodic scalar conservation laws
SIAM Journal on Numerical Analysis
Shock capturing by the spectral viscosity method
ICOSAHOM '89 Proceedings of the conference on Spectral and high order methods for partial differential equations
Legendre pseudospectral viscosity method for nonlinear conservation laws
SIAM Journal on Numerical Analysis
A note on the accuracy of spectral method applied to nonlinear conservation laws
Journal of Scientific Computing
Enhanced spectral viscosity approximations for conservation laws
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
Spectral Vanishing Viscosity Method For Nonlinear Conservation Laws
SIAM Journal on Numerical Analysis
Detection of Edges in Spectral Data II. Nonlinear Enhancement
SIAM Journal on Numerical Analysis
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Spectral approximations to nonlinear hyperbolic conservation laws require dissipative regularization 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 particular, 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.