Structure-preserving finite element discretization and optimal control of the shallow water equations with bathymetry on unstructured meshes

该项目的目标是将一族现代代数通量校正格式推广到带源项的浅水方程(SWES)。此外，我们将开发一个基于优化的工具，用于根据自由表面高程的实验数据重建水深测量(海底地形)。我们使用连续有限元对控制方程进行离散，并对标准Galerkin近似进行修正，以确保所有相关约束(水高的非负性、水高和流速分量的局部极大值原理、熵不等、与稳态平衡的一致性)的有效性。我们的半离散格式的保界通量限制器的推导是基于凸分析和关于允许的中间态的表示。利用反扩散通量的熵产生速率的限制器来加强熵的稳定性。非结构化网格和隐式时间积分器的使用是可能的。对SWES的良好平衡扩展需要对齐次双曲型系统的代数限制技术进行仔细且在理论上合理的调整。将特别注意干态和干湿转变的数值处理。未知的水深测量将通过求解优化问题来重构，其中SWE系统和反问题的连续性方程作为PDE约束。最优离散拉普拉斯控制将取代我们以前项目相关工作中使用的人工正则化项。最优控制问题的新表述以及解决这些问题的方法，使我们的方法有别于传统方法。软件开发将在开源的C++库https://mfem.org)，的基础上进行，我们将为其贡献一个有限元工具箱，用于基于有限元分析的地球物理流动模拟。

The objective of this project is to extend a family of modern algebraic flux correction schemes to the shallow water equations (SWEs) with source terms. Additionally, we will develop an optimization-based tool for reconstruction of bathymetry (bottom topography) from experimental data for the free surface elevation. We discretize the governing equations using continuous finite elements and modify the standard Galerkin approximation in a way which provably guarantees the validity of all relevant constraints (nonnegativity of the water height, local maximum principles for the water height and velocity components, entropy inequalities, consistency with steady-state equilibria). The derivation of bound-preserving flux limiters for our semi-discrete schemes is based on convex analysis and representations in terms of admissible intermediate states. Entropy stability is enforced using a limiter for the rate of entropy production by antidiffusive fluxes. The use of unstructured meshes and implicit time integrators is possible. Well-balanced extensions to SWEs require a careful and theoretically justified adaptation of our algebraic limiting techniques for homogeneous hyperbolic systems. Special attention will be paid to the numerical treatment of dry states and wet-dry transitions. Unknown bathymetry will be reconstructed by solving optimization problems, in which the SWE system and the continuity equation of an inverse problem serve as PDE constraints. An optimal discrete Laplacian control will replace an artificial regularization term that was used in our previous project-related work. The novel formulation of optimal control problems, and the way in which they are solved, distinguishes our method from conventional approaches. Software development will be performed on the basis of the open-source C++ library MFEM (https://mfem.org), to which we will contribute a finite element toolbox for SWE-based simulations of geophysical flows.