Spectral Viscosity Method with Generalized Hermite Functions and Its Spectral Filter for Nonlinear Conservation Laws

In this paper, we propose a new spectral viscosity method for the solution of nonlinear scalar conservation laws in the whole line. The proposed method is based on the generalized Hermite functions. It is shown rigorously that this scheme converges to the unique entropy solution by using compensated compactness arguments. The numerical experiments of the inviscid Burger’s equation support our result. Moreover, we step further to discuss the postprocessing method of the approximate solution obtained by our method to eliminate the oscillation away from the discontinuity. We suggest a new spectral filter defined by a kernel function in the physical space, and show the spectral accuracy at the points away from the discontinuity.