6 research outputs found
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 ,
where 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