Spectral Vanishing Viscosity Method For Nonlinear Conservation Laws

We propose a new spectral viscosity (SV) scheme for the accurate solution of nonlinear conservation laws. It is proved that the SV solution converges to the unique entropy solution under appropriate reasonable conditions. The proposed SV scheme is implemented directly on high modes of the computed solution. This should be compared with the original nonperiodic SV scheme introduced by Maday, Ould Kaber, and Tadmor in [SIAM J. Numer. Anal., 30 (1993), 321--342], where SV is activated on the derivative of the SV solution. The new proposed SV method could be viewed as a correction of the former, and it offers an improvement which is confirmed by our numerical experiments. A postprocessing method is implemented to greatly enhance the accuracy of the computed SV solution. The numerical results show the efficiency of the new method.