  Combining the characteristic method and the local discontinuous Galerkin
method with carefully constructing numerical fluxes, we design the variational
formulations for the time-dependent convection-dominated Navier-Stokes
equations in $\mathbb{R}^2$. The proposed symmetric variational formulation is
strictly proved to be unconditionally stable; and the scheme has the striking
benefit that the conditional number of the matrix of the corresponding matrix
equation does not increase with the refining of the meshes. The presented
scheme works well for a wide range of Reynolds numbers, e.g., the scheme still
has good error convergence when $Re=0.5 e+005$ or $1.0 e+ 008$. Extensive
numerical experiments are performed to show the optimal convergence orders and
the contours of the solutions of the equation with given initial and boundary
conditions.
