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We present a new approach to stabilizing high-order Runge-Kutta discontinuousGalerkin (RKDG) schemes using weighted essentially non-oscillatory (WENO)reconstructions in the context of hyperbolic conservation laws. In contrast toRKDG schemes that overwrite finite element solutions with WENO reconstructions,our approach employs the reconstruction-based smoothness sensor presented byKuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) to control the amount ofadded numerical dissipation. Incorporating a dissipation-based WENOstabilization term into a discontinuous Galerkin (DG) discretization, theproposed methodology achieves high-order accuracy while effectively capturingdiscontinuities in the solution. As such, our approach offers an attractivealternative to WENO-based slope limiters for DG schemes. The reconstructionprocedure that we use performs Hermite interpolation on stencils composed of amesh cell and its neighboring cells. The amount of numerical dissipation isdetermined by the relative differences between the partial derivatives ofreconstructed candidate polynomials and those of the underlying finite elementapproximation. The employed smoothness sensor takes all derivatives intoaccount to properly assess the local smoothness of a high-order DG solution.Numerical experiments demonstrate the ability of our scheme to capturediscontinuities sharply. Optimal convergence rates are obtained for allpolynomial degrees.

Dissipative WENO stabilization of high-order discontinuous Galerkin methods for hyperbolic problems