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