Dissipation-based WENO stabilization of high-order finite element methods for scalar conservation laws

We present a new perspective on the use of weighted essentially nonoscillatory (WENO) reconstructions in high-order methods for scalar hyperbolic conservation laws. The main focus of this work is on nonlinear stabilization of continuous Galerkin (CG) approximations. The proposed methodology also provides an interesting alternative to WENO-based limiters for discontinuous Galerkin (DG) methods. Unlike Runge--Kutta DG schemes that overwrite finite element solutions with WENO reconstructions, our approach uses a reconstruction-based smoothness sensor to blend the numerical viscosity operators of high- and low-order stabilization terms. The so-defined WENO approximation introduces low-order nonlinear diffusion in the vicinity of shocks, while preserving the high-order accuracy of a linearly stable baseline discretization in regions where the exact solution is sufficiently smooth. The underlying reconstruction procedure performs Hermite interpolation on stencils consisting of a mesh cell and its neighbors. The amount of numerical dissipation depends on the relative differences between partial derivatives of reconstructed candidate polynomials and those of the underlying finite element approximation. All derivatives are taken into account by the employed smoothness sensor. To assess the accuracy of our CG-WENO scheme, we derive error estimates and perform numerical experiments. In particular, we prove that the consistency error of the nonlinear stabilization is of the order $p+1/2$, where $p$ is the polynomial degree. This estimate is optimal for general meshes. For uniform meshes and smooth exact solutions, the experimentally observed rate of convergence is as high as $p+1$.