Chebyshev-Legendre super spectral viscosity method for nonlinear conservation laws

In this paper, a super spectral viscosity method using the Chebyshev differential operator of high order D-s = (root 1-x(2) partial derivative(x))(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.