We propose a way to maintain strong consistency and facilitate error analysis in the context of dissipation-based WENO stabilization for continuous and discontinuous Galerkin discretizations of conservation laws. Following Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) and Vedral (arXiv preprint arXiv:2309.12019), we use WENO shock detectors to determine appropriate amounts of low-order artificial viscosity. In contrast to existing WENO methods, our approach blends candidate polynomials using residual-based nonlinear weights. The shock-capturing terms of our stabilized Galerkin methods vanish if residuals do. This enables us to achieve improved accuracy compared to weakly consistent alternatives. As we show in the context of steady convection-diffusion-reaction (CDR) equations, nonlinear local projection stabilization terms can be included in a way that preserves the coercivity of local bilinear forms. For the corresponding Galerkin-WENO discretization of a CDR problem, we rigorously derive a priori error estimates. Additionally, we demonstrate the stability and accuracy of the proposed method through one- and two-dimensional numerical experiments for hyperbolic conservation laws and systems thereof. The numerical results for representative test problems are superior to those obtained with traditional WENO schemes, particularly in scenarios involving shocks and steep gradients.

翻译：针对守恒律的连续及间断Galerkin离散格式，本文提出一种在基于耗散的WENO稳定化框架中保持强一致性并便于误差分析的方法。遵循Kuzmin与Vedral（J. Comput. Phys. 487:112153, 2023）及Vedral（arXiv预印本arXiv:2309.12019）的研究思路，我们采用WENO激波探测器来确定恰当的低阶人工黏性量。与现有WENO方法不同，本方法采用基于残差的非线性权值对候选多项式进行混合。当残差为零时，稳定化Galerkin方法的激波捕捉项亦随之消失，这使得我们能够获得相较于弱一致性方法更高的精度。如我们在稳态对流-扩散-反应方程背景下所论证的，非线性局部投影稳定项可以融入系统，同时保持局部双线性形式的强制性。针对对流-扩散-反应问题的Galerkin-WENO离散格式，我们严格推导了先验误差估计。此外，通过对双曲守恒律及其方程组开展一维与二维数值实验，验证了所提方法的稳定性与精度。代表性测试问题的数值结果优于传统WENO格式，在涉及激波与陡峭梯度的场景中表现尤为突出。