  In this paper, we propose new spectral viscosity methods based on the
generalized Hermite functions for the solution of nonlinear scalar conservation
laws in the whole line. It is shown rigorously that these schemes converge to
the unique entropy solution by using compensated compactness arguments, under
some conditions. The numerical experiments of the inviscid Burger's equation
support our result, and it verifies the reasonableness of the conditions.
