Adaptive Spectral Viscosity for Hyperbolic Conservation Laws

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.