Chebyshev--Legendre Super Spectral Viscosity Method for Nonlinear Conservation Laws

Abstract

In this paper, a super spectral viscosity method using the Chebyshev differential operator of high order $D^s=(\sqrt{1-x^2} \opx)^s $ is developed for nonlinear conservation laws. The boundary conditions are treated by a penalty method. Compared with the second-order spectral viscosity method, the super one is much weaker while still guaranteeing the convergence of the bounded solution of the Chebyshev--Galerkin, Chebyshev collocation, or Legendre--Galerkin approximations to nonlinear conservation laws, which is proved by compensated compactness arguments.