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, split forms are examined as they enable 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 value c as that of the one-dimensional ESFR scheme.
翻译:具体地说,能源稳定通量重建(ESFR)计划被认为对于在曲线坐标上制定稳定DG计划至关重要,因为这种计划允许在曲线坐标上为线性对冲问题设计灵活度和稳定证明;此外,对不同的形式进行了研究,因为它们有助于制定能源稳定性证明;第一个关键步骤证明,在曲线坐标上,不连续的Galerkin(DG)保守和非保守形式本质上是不同的,即使是在精确的整合和分析精确的衡量术语下。这一分析表明,分形形式对于在曲线坐标上制定稳定DG计划至关重要,并且鼓励在每一个元素上构建依赖于线性对线性对冲问题的纠正功能。此外,可证实稳定的FR计划不同于文献中仅将ESFR校正功能应用于表面或保守形式的计划,而是将ESFR校正功能纳入方程式的完全分解形式。 事实证明,当校正功能仅用于草底线坐标对等值的地面重建时,这一计划就不同了。最后,我们从数字上核查了拟议的ESFR规则的稳定性主张,以新的FRFR格式显示我们的拟议格式。