非线性稳定的分裂形式通量重构高阶方法
Nonlinearly Stable Flux Reconstruction High-Order Methods in Split Form
摘要：
通量重建（FR）方法在研究界得到了普及，因为它通过模态滤波校正场恢复有前途的高阶方法，例如在复杂几何形状的非结构化网格上的间断Galerkin方法等。此外，FR计划，特别是能量稳定FR（ESFR）计划，也被称为convention-Castonguay-Jameson-Huynh计划，已被证明是有吸引力的，因为它们允许设计灵活性以及仿射元素上的线性平流问题的稳定性证明。此外，分裂形式最近看到了复苏的研究活动，由于其所得的非线性（熵）稳定性证明。本文首次推导出分裂形式的非线性稳定ESFR格式，该格式能够证明非配置模态ESFR分裂形式具有不同的体积和表面体积节点的非线性稳定性。关键使能技术是将分裂应用于离散刚度算子。这自然会导致适当的表面和数值通量，使熵稳定性和守恒证明。当这些计划重铸强的形式，它们不同于计划中发现的ESFR文献中的ESFR修正功能纳入体积积分。此外，数值实验进行验证，新的一类建议ESFR分裂形式是非线性稳定的对比标准分裂形式ESFR方法。最后，新的ESFR分裂形式被证明可以获得正确的精度阶数。
英文摘要：
The flux reconstruction (FR) method has gained popularity in the research community as it recovers promising high-order methods through modally filtered correction fields, such as the discontinuous Galerkin method, amongst others, on unstructured grids over complex geometries. Moreover, FR schemes, specifically energy stable FR (ESFR) schemes also known as Vincent-Castonguay-Jameson-Huynh schemes, have proven attractive as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms have recently seen a resurgence in research activity due to their resultant nonlinear (entropy) stability proofs. This paper derives for the first time nonlinearly stable ESFR schemes in split form that enable nonlinear stability proofs for, uncollocated, modal, ESFR split forms with different volume and surface cubature nodes. The critical enabling technology is applying the splitting to the discrete stiffness operator. This naturally leads to appropriate surface and numerical fluxes, enabling both entropy stability and conservation proofs. When these schemes are recast in strong form, they differ from schemes found in the ESFR literature as the ESFR correction functions are incorporated on the volume integral. Furthermore, numerical experiments are conducted verifying that the new class of proposed ESFR split forms is nonlinearly stable in contrast to the standard split form ESFR approach. Lastly, the new ESFR split form is shown to obtain the correct orders of accuracy.
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