In this work, in a monodimensional setting, 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. Despite the simulations addressing the monodimensional SW equations only, many elements can be extended to other general hyperbolic systems and to a multidimensional setting.

翻译：本研究在一维框架下，研究了浅水方程新型连续内部惩罚（CIP）稳定化的高阶精度与平衡（WB）特性。底层任意高阶数值框架采用残差分布（RD）/连续伽辽金（CG）有限元方法（FEM）进行空间离散，结合延迟校正（DeC）时间积分方法，构建全显式格式。一方面，所引入的CIP稳定化方法均经专门设计以保证精确维持静水稳态；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，其显式解析表达式未知。数值实验考虑了多种基函数，所有算例均验证了新型稳定化方法的高阶精度与精确维持静水稳态的能力，并可捕捉该平衡态的小扰动。此外，基于空间残差与全局通量概念的若干方法，在封闭形式未知的一般稳态解背景下展现出优异性能与超收敛特性。尽管仅针对一维浅水方程进行模拟，但多数要素可拓展至其他双曲系统及多维框架。