Enriched Galerkin method for the shallow-water equations
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Enriched Galerkin method for the shallow-water equations Moritz Hauck1 · Vadym Aizinger2 Andreas Rupp4
· Florian Frank1
· Hennes Hajduk3 ·
Received: 29 April 2020 / Accepted: 12 November 2020 / Published online: 21 November 2020 © The Author(s) 2020
Abstract This work presents an enriched Galerkin (EG) discretization for the two-dimensional shallow-water equations. The EG finite element spaces are obtained by extending the approximation spaces of the classical finite elements by discontinuous functions supported on elements. The simplest EG space is constructed by enriching the piecewise linear continuous Galerkin space with discontinuous, element-wise constant functions. Similar to discontinuous Galerkin (DG) discretizations, the EG scheme is locally conservative, while, in multiple space dimensions, the EG space is significantly smaller than that of the DG method. This implies a lower number of degrees of freedom compared to the DG method. The EG discretization presented for the shallow-water equations is well-balanced, in the sense that it preserves lake-at-rest configurations. We evaluate the method’s robustness and accuracy using various analytical and realistic problems and compare the results to those obtained using the DG method. Finally, we briefly discuss implementation aspects of the EG method within our MATLAB / GNU Octave framework FESTUNG. Keywords Enriched Galerkin · Finite elements · Shallow-water equations · Discontinuous Galerkin · Local conservation · Ocean modeling Mathematics Subject Classification 65 · 76 · 86
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Vadym Aizinger [email protected]
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Chair for Computational Mathematics, University of Augsburg, Universitätsstraße 2, 86159 Augsburg, Germany
2
Chair of Scientific Computing, University of Bayreuth, Universitätsstraße 30, 95447 Bayreuth, Germany
3
Institute of Applied Mathematics (LS III), TU Dortmund University, Vogelpothsweg 87, 44227 Dortmund, Germany
4
Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
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GEM - International Journal on Geomathematics (2020) 11:31
1 Introduction The two dimensional shallow-water equations (SWE) are used for a wide range of applications in environmental and hydraulic engineering, oceanography, and many other areas. They are discretized on computational domains that can be very large and often feature complex geometries; therefore, the numerical schemes must be computationally efficient and robust. The nonlinearity and hyperbolic character of the SWE system constitute additional challenges for designing discretizations and solution algorithms, while other application-specific aspects such as local conservation of unknown quantities and well-balancedness represent further desirable properties [see Hinkelmann et al. (2015)] for a brief overview of key requirements for SWE models). The aforementioned issues led to a large number of studies dedicated to the development, analysis, and practical evaluation of various numerical te
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