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

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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.