We present a new approach to stabilizing high-order Runge-Kutta discontinuous Galerkin (RKDG) schemes using weighted essentially non-oscillatory (WENO) reconstructions in the context of hyperbolic conservation laws. In contrast to RKDG schemes that overwrite finite element solutions with WENO reconstructions, our approach employs the reconstruction-based smoothness sensor presented by Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) to control the amount of added numerical dissipation. Incorporating a dissipation-based WENO stabilization term into a discontinuous Galerkin (DG) discretization, the proposed methodology achieves high-order accuracy while effectively capturing discontinuities in the solution. As such, our approach offers an attractive alternative to WENO-based slope limiters for DG schemes. The reconstruction procedure that we use performs Hermite interpolation on stencils composed of a mesh cell and its neighboring cells. The amount of numerical dissipation is determined by the relative differences between the partial derivatives of reconstructed candidate polynomials and those of the underlying finite element approximation. The employed smoothness sensor takes all derivatives into account to properly assess the local smoothness of a high-order DG solution. Numerical experiments demonstrate the ability of our scheme to capture discontinuities sharply. Optimal convergence rates are obtained for all polynomial degrees.

翻译：我们提出了一种新方法，通过使用加权本质无振荡（WENO）重构来稳定高阶龙格-库塔间断伽辽金（RKDG）格式，应用于双曲守恒律。与用WENO重构覆盖有限元解的RKDG格式不同，我们的方法采用Kuzmin和Vedral（J. Comput. Phys. 487:112153, 2023）提出的基于重构的光滑度传感器来控制添加的数值耗散量。通过将基于耗散的WENO稳定项融入间断伽辽金（DG）离散化中，所提出的方法在有效捕获解的不连续性的同时实现了高阶精度。因此，我们的方法为DG格式中基于WENO的斜率限制器提供了一种有吸引力的替代方案。我们使用的重构过程在由网格单元及其相邻单元组成的模板上执行Hermite插值。数值耗散量由重构候选多项式偏导数与底层有限元近似偏导数之间的相对差异决定。所采用的光滑度传感器考虑所有导数，以正确评估高阶DG解的局部光滑性。数值实验证明了我们的格式能够尖锐地捕获不连续性。对于所有多项式次数，均获得了最优收敛速率。