Workshop on Numerical Methods and Analysis in CFD - Abstract

Ruppenthal, Falko

Bathymetry reconstruction via optimal control in well-balanced finite element methods for the shallow water equations

Co-author: D. Kuzmin (Technische Universität Dortmund)

The reconstruction of bathymetry from known free surface elevation via numerical solution of the shallow water equations (SWE) is an ill-conditioned and generally ill--posed inverse problem. Without proper regularization, a method may fail to converge, or steady-state topography may exhibit spurious oscillations even if a smooth exact solution is known to exist. The presence of noise in the data, or a poor choice of discretization techniques aggravates such issues. To filter out perturbations caused by ill-conditioning and/or presence of unresolvable fine-scale features, a numerical method for the inverse SWE problem must be equipped with carefully designed stabilization operators [2].

In this work, we discretize the two-dimensional shallow water equations using continuous linear finite elements. The current implementation uses a new well-balanced extension of the algebraic Lax--Friedrichs method analyzed in [1] to the SWE system with non-flat topography. The steady-state bathymetry is calculated via time marching. Two approaches are used to cure the lack of well-posedness and avoid oscillations. The first one adds an artificial diffusion term to the conservation law for the water height (as in [2]). In the second approach, the regularization term consists of numerical fluxes that are constructed using a new optimal control method. An optimization problem is formulated for scalar flux potentials with the aim of minimizing the perturbation of the discretized shallow water equations and deviations from the measured free surface elevation. To suppress oscillations caused by non-smooth data, we use the total variation denoising approach developed in [3]. The first numerical results for one- and two-dimensional test problems are promising. These preliminary results include convergence studies for benchmarks with noise in the initial data and experiments with discontinuous bathymetry.

References:
[1] J.-L. Guermond and B. Popov, Invariant domains and first-order continuous finite element approximation for hyperbolic systems, SIAM Journal on Numererical Analysis, 54 (2016), pp. 2466--2489.
[2] H. Hajduk, D. Kuzmin, and V. Aizinger, Bathymetry reconstruction using inverse shallow water models: Finite element discretization and regularization, Chapter 20 in: Van Brummelen et al. (eds), Numerical Methods for Flows<, LNCSE, Springer, (2020), pp. 223--230.
[3] R. Vogel, and M. E. Oman, Iterative methods for total variation denoising, SIAM Journal on Scientific Computing, 17:1 (1996), pp. 227--238.