Enhanced spectral viscosity approximations for conservation laws

In this paper we construct, analyze and implement a new procedure for the spectral approximations of nonlinear conservation laws. It is well known that using spectral methods for nonlinear conservation laws will result in the formation of the Gibbs phenomenon once spontaneous shock discontinuities appear in the solution. These spurious oscillations will in turn lead to loss of resolution and render the standard spectral approximations unstable. The Spectral Viscosity (SV-) method (Tadmor, 1989) was developed to stabilize the spectral method by adding a spectrally small amount of high-frequencies diffusion carried out in the dual space. The resulting SV-approximation is stable without sacrificing spectral accuracy. The SV-method recovers a spectrally accurate approximation to the projection of the entropy solution; the exact projection, however, is at best a first order approximation to the exact solution as a result of the formation of the shock discontinuities. The issue of spectral resolution is addressed by post-processing the SV-solution to remove the spurious oscillations at the discontinuities, as well as increase the first-order - O(1/N) accuracy away from the shock discontinuities. Successful post-processing methods have been developed to eliminate the Gibbs phenomenon and recover spectral accuracy for the SV-approximation. However, such reconstruction methods require a priori knowledge of the locations of the shock discontinuities. Therefore, the detection of these discontinuities is essential to obtain an overall spectrally accurate solution. To this end, we employ the recently constructed enhanced edge detectors based on appropriate concentration factors (Gelb and Tadmor, 1999). Once the edges of these discontinuities are identified, we can utilize a post-processing reconstruction method, and show that the post-processed SV-solution recovers the exact entropy solution with remarkably high-resolution. We apply our new numerical method, the Enhanced SV-method, to two numerical examples, the scalar periodic Burgers' equation and the one-dimensional system of Euler equations of gas dynamics. Both approximations exhibit high accuracy and resolution to the exact entropy solution.