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

本计画的目的是将现代代数通量修正格式推广至含源项的浅水方程。此外，我们将开发一种基于优化的工具，用于从自由表面高程的实验数据重建水深（海底地形）。我们离散的控制方程，使用连续有限元和修改的标准Galerkin近似的方式，可证明保证所有相关的约束（非负的水的高度，局部最大值原则的水的高度和速度分量，熵的不平等，与稳态平衡的一致性）的有效性。我们的半离散方案的保界通量限制器的推导是基于凸分析和表示的容许的中间状态。熵的稳定性是强制执行使用的限制器的熵产生率的反扩散通量。使用非结构网格和隐式时间积分器是可能的。SWES的平衡扩展需要仔细和理论上合理的适应我们的代数限制齐次双曲系统的技术。将特别注意干燥状态和湿-干转换的数值处理。以SWE系统和反问题的连续性方程作为PDE约束，通过求解优化问题重构未知水深。最佳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。最优控制问题的新提法，以及它们的解决方式，将我们的方法与传统方法区分开来。软件开发将在开源C++库MFEM（https：//www.example.com）的基础上进行，我们将为基于SWE的地球物理流模拟提供有限元工具箱。mfem.org

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.