松弛延迟修正法及其在残差分布格式中的应用
Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes
摘要：
在文献[1]中提出了一种简化的DeC方法，当与残差分布（RD）框架相结合时，它允许构造一个高阶、显式的有限元格式，具有连续逼近，避免了双曲问题的质量矩阵的反演。在本文中，我们关闭一些开放的差距，在延迟校正（DeC）的背景下，其应用范围内的RD框架。首先，我们证明了DeC计划和RK方法之间的联系。有了这些知识，DeC可以重写为显式欧拉步骤的凸组合，显示与强稳定性保持（SSP）框架的连接。然后，我们可以应用[2]中介绍的松弛方法，并使用[3]中提出的熵校正函数构造熵守恒/耗散DeC（RDeC）方法。[1]R.阿布格罗使用全局连续逼近和避免质量矩阵的双曲型问题的高阶格式。Journal of Scientific Computing，73（2）：461 - 494，Dec 2017. [2]D.凯奇森松弛龙格-库塔法：内积范数的守恒性和稳定性。SIAM Journal on数值分析，57（6）：2850 - 2870，2019。[3]R.阿布格罗构造满足附加守恒关系的格式的一般框架。应用于熵守恒和熵耗散格式。计算物理学杂志，372：640 - 666，2018。
英文摘要：
In [1] is proposed a simplified DeC method, that, when combined with the residual distribution (RD) framework, allows to construct a high order, explicit FE scheme with continuous approximation avoiding the inversion of the mass matrix for hyperbolic problems. In this paper, we close some open gaps in the context of deferred correction (DeC) and their application within the RD framework. First, we demonstrate the connection between the DeC schemes and the RK methods. With this knowledge, DeC can be rewritten as a convex combination of explicit Euler steps, showing the connection to the strong stability preserving (SSP) framework. Then, we can apply the relaxation approach introduced in [2] and construct entropy conservative/dissipative DeC (RDeC) methods, using the entropy correction function proposed in [3]. [1] R. Abgrall. High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. Journal of Scientific Computing, 73(2):461--494, Dec 2017. [2] D. Ketcheson. Relaxation Runge--Kutta methods: Conservation and stability for inner-product norms. SIAM Journal on Numerical Analysis, 57(6):2850--2870, 2019. [3] R. Abgrall. A general framework to construct schemes satisfying additional conservation relations. application to entropy conservative and entropy dissipative schemes. Journal of Computational Physics, 372:640--666, 2018.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
