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.
Highlights
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
A big advantage of the finite element approach is its potential to naturally accommodate higher-order discretizations on unstructured meshes; in this vein, various methods based on the continuous Galerkin (CG) and discontinuous Galerkin (DG) approximation spaces have been described and compared in the literature (Hanert et al 2003; Comblen et al 2010)
To give an indication of the computational costs associated with an enriched Galerkin (EG) scheme, we list in Table 3 the cumulative solution times for systems with the DG mass matrix (P1,1), the EG mass matrix (P1,0), and the EG mass matrix (P1,0) lumped according to the scheme proposed in Becker et al (2003)
Summary
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. A big advantage of the finite element approach is its potential to naturally accommodate higher-order discretizations on unstructured meshes; in this vein, various methods based on the continuous Galerkin (CG) and discontinuous Galerkin (DG) approximation spaces (or mixtures of both) have been described and compared in the literature (Hanert et al 2003; Comblen et al 2010). The results of these comparisons can be summarized (in a somewhat oversimplified fashion) as follows:. The method is implemented in the FESTUNG framework (Frank et al 2015; Reuter et al 2016; Jaust et al 2018; Reuter et al 2020) by modifying our DG implementation for the
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