Provably stable flux reconstruction high-order methods on curvilinear elements
Provably stable flux reconstruction high-order methods on curvilinear elements
复制标题
曲线元素上可证明稳定的通量重建高阶方法
DOI：
10.1016/j.jcp.2022.111259
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发表时间：
2022
影响因子：
4.1
通讯作者：
Carpenter, Mark H.
中科院分区：
文献类型：
--
作者：
Cicchino, Alexander;Del Rey Fernández, David C.;Nadarajah, Siva;Chan, Jesse;Carpenter, Mark H.
对曲线坐标系下的偏微分方程导出了可证稳定的通量重建（FR）格式。具体而言，能量稳定通量重建（ESFR）计划被认为是因为它们允许设计的灵活性，以及仿射元素的线性平流问题的稳定性证明。此外，曲线度量分裂形式的线性物理通量进行检查，因为它使能源稳定性证明的发展。第一个关键步骤证明，在曲线坐标系中，间断Galerkin（DG）保守和非保守形式是固有的不同，即使在精确积分和解析精确度量条款。该分析表明，分裂形式是必不可少的曲线坐标上的可证明稳定的DG计划，并激励建设度量相关的ESFR校正功能，在每个元素。此外，可证明稳定的FR格式不同于文献中仅将ESFR校正函数应用于表面项或保守形式的格式，而是将ESFR校正函数应用于方程的完全分裂形式。当修正函数仅用于曲线坐标系下的曲面重构时，该格式是发散的。我们数值验证了我们提出的FR分裂形式的稳定性声明，并将其与文献中的ESFR方案进行比较。最后，新提出的可证明稳定的FR计划，以获得最佳的收敛阶。该格式在与一维ESFR格式相当的修正参数值下失去了几个精度阶。
Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, the curvilinear metric split-form for a linear physical flux is examined as it enables the development of energy stability proofs. The first critical step proves, that in curvilinear coordinates, the discontinuous Galerkin (DG) conservative and non-conservative forms are inherently different–even under exact integration and analytically exact metric terms. This analysis demonstrates that the split form is essential to developing provably stable DG schemes on curvilinear coordinates and motivates the construction of metric dependent ESFR correction functions in each element. Furthermore, the provably stable FR schemes differ from schemes in the literature that only apply the ESFR correction functions to surface terms or on the conservative form, and instead incorporate the ESFR correction functions on the full split form of the equations. It is demonstrated that the scheme is divergent when the correction functions are only used for surface reconstruction in curvilinear coordinates. We numerically verify the stability claims for our proposed FR split forms and compare them to ESFR schemes in the literature. Lastly, the newly proposed provably stable FR schemes are shown to obtain optimal orders of convergence. The scheme loses the orders of accuracy at the equivalent correction parameter valuecas that of the one-dimensional ESFR scheme.