In this work, the high order accuracy and the well-balanced (WB) properties of some novel continuous interior penalty (CIP) stabilizations for the Shallow Water (SW) equations are investigated. The underlying arbitrary high order numerical framework is given by a Residual Distribution (RD)/continuous Galerkin (CG) finite element method (FEM) setting for the space discretization coupled with a Deferred Correction (DeC) time integration, to have a fully-explicit scheme. If, on the one hand, the introduced CIP stabilizations are all specifically designed to guarantee the exact preservation of the lake at rest steady state, on the other hand, some of them make use of general structures to tackle the preservation of general steady states, whose explicit analytical expression is not known. Several basis functions have been considered in the numerical experiments and, in all cases, the numerical results confirm the high order accuracy and the ability of the novel stabilizations to exactly preserve the lake at rest steady state and to capture small perturbations of such equilibrium. Moreover, some of them, based on the notions of space residual and global flux, have shown very good performances and superconvergences in the context of general steady solutions not known in closed-form. Many elements introduced here can be extended to other hyperbolic systems, e.g., to the Euler equations with gravity.

翻译：本文研究了针对浅水方程的新型连续内部罚项（CIP）稳定化方法的高阶精度与平衡（WB）特性。该方法的任意高阶数值框架基于残差分布（RD）/连续伽辽金（CG）有限元法（FEM）进行空间离散，并结合延迟校正（DeC）时间积分，构成全显式格式。一方面，所引入的CIP稳定化方法均专门设计以确保静水稳态的精确保持；另一方面，部分方法采用通用结构处理一般稳态的保持问题，而此类稳态通常缺乏显式解析表达式。数值实验中考虑了多种基函数，在所有情形下，数值结果均验证了新稳定化方法的高阶精度、静水稳态的精确保持能力以及对稳态小扰动的捕捉能力。此外，基于空间残差与全局通量概念的若干方法在未知闭式解的一般稳态问题中展现出优异的性能与超收敛特性。本文提出的多项技术可推广至其他双曲系统，例如含重力项的欧拉方程。