分裂格式对间断伽略金（DG）格式的混叠误差与熵稳定性的作用研究
On the Role of Split Formulations on Aliasing Errors and Entropy Stability of Discontinuous Galerkin Schemes
AI总结：
本文研究分裂格式DG格式对一维Burgers问题的作用，推广谱离散化的去混叠证明，确定优化分裂系数，阐明DG混叠误差与熵稳定性的联系，证明熵稳定分裂可降低弱欠分辨范围的混叠误差并维持解的长期稳定性。
AI中文摘要：
在本研究中，我们针对一维Burgers问题，正式研究了分裂格式间断伽略金（DG）离散化的去混叠特性。通过推广Blaisdell等人针对谱离散化的证明，我们表明分裂格式DG格式在弱欠分辨范围内，可通过守恒型与非守恒型形式的积分误差抵消实现去混叠。作为推论，我们确定了与求积规则和阶数相关的分裂系数对，这些系数可消除混叠误差的主导分量。尽管这些“优化”的分裂系数可最小化数值格式的积分误差，但我们证明，由熵稳定分裂产生的斜对称混叠模式是维持数值解长期稳定性的必要条件。所提出的框架为Gassner提出的(1/3, 2/3)分裂格式的DG离散化的熵稳定性提供了替代证明，阐明了DG混叠误差与熵稳定性之间的联系。最后，我们还表明，与守恒型形式相比，熵稳定分裂在弱欠分辨范围内与更低的混叠误差相关。
英文摘要：
In this work, we formally investigate the dealiasing properties of split form discontinuous Galerkin (DG) discretizations for the one-dimensional Burgers problem. By generalizing the proof of Blaisdell et al. for spectral discretizations, we show that split form DG schemes achieve dealiasing in the weakly underresolved range through integration error cancellation on the conservative and non-conservative forms. As a corollary, we identify quadrature- and order-dependent pairs of splitting coefficients that eliminate the dominant component of the aliasing error. While these ``optimized'' splitting coefficients minimize integration errors on the numerical scheme, we show that the skew-symmetric aliasing pattern resulting from the entropy stable split is required to maintain long-term stability of the numerical solution. The proposed framework provides an alternative proof for the entropy stability of the DG discretization of the (1/3, 2/3) split formulation introduced by Gassner which clarifies the connection between DG aliasing errors and entropy stability. Finally, we also show that the entropy stable split is associated with lower aliasing errors in the weakly underresolved range compared to the conservative form.
