Chebyshev–legendre Spectral Viscosity Method for Nonlinear Conservation Laws∗

In this paper, a Chebyshev–Legendre spectral viscosity (CLSV) method is developed for nonlinear conservation laws with initial and boundary conditions. The boundary conditions are dealt with by a penalty method. The viscosity is put only on the high modes, so accuracy may be recovered by postprocessing the CLSV approximation. It is proved that the bounded solution of the CLSV method converges to the exact scalar entropy solution by compensated compactness arguments. Also, a new spectral viscosity method using the Chebyshev differential operator D = √ 1− x2∂x is introduced, which is a little weaker than the usual one while guaranteeing the convergence of the bounded solution of the Chebyshev Galerkin, Chebyshev collocation, or Legendre Galerkin approximation to nonlinear conservation laws. This kind of viscosity is ready to be generalized to a super viscosity version.