Raviart–Thomas and Brezzi–Douglas–Marini finite‐element approximations of the shallow‐water equations

Abstract

SUMMARYAn analysis of the discrete shallow-water equations using the Raviart-Thomas and Brezzi-Douglas-Marini finite elements is presented. For inertia-gravity waves, the discrete formulations are obtained and the dispersion relations are computed in order to quantify the dispersive nature of the schemes on two meshes made up of equilateral and biased triangles. A linear algebra approach is also used to ascertain the possible presence of spurious modes arising from the discretization. The geostrophic balance is examined and the smallest representable vortices are characterized on both structured and unstructured meshes. Numerical solutions of two test problems to simulate gravity and Rossby modes are in good agreement with the analytical results.

Spurious inertial oscillations in shallow-water models

“…The dispersion relations obtained for the P DG 1 -P 1 and P DG 1 -P 2 pairs are supplemented with existing results performed in [44,58] and rearranged in Table 3, in order to better understand the origin of the spurious inertial modes, and how they are distinguished from other types of modes. Two groups of FE pairs, named 1 and 2, are considered in Table 3.…”

“…In order to compute the dispersion relations for the FE pairs of each group, we proceed as in Sections 3.2 and 3.3. An appropriate variational formulation, namely (13) and (14) or (15) and (16), is employed where the spatially discrete operators are obtained from the stencils given in [44,58]. The choice of the variational formulation depends on the regularity of the velocity and surface-elevation fields.…”

“…Finally, the BDM 1 element [11,58] can be seen as an enriched version of RT 0 . The normal component is still continuous through triangle edges, as for the RT 0 element, but the corresponding basis functions are linear instead of being constant.…”

“…However, the multiplicity of the spurious inertial modes x = ±f is not mentioned in [18], although this is crucial to understand the behaviour of such modes, as shown in Section 5. Further, as observed in [5,25,46,58] the shape and configuration of the triangles affect the accuracy of the FE implementation, and the use of equilateral triangles diminishes the directional dependence of the solutions. Numerical experiments have also demonstrated that equilateral triangles are more accurate that right triangles [31].…”

A New Mathematical Model of Heat Equations And its Application on the Agriculture Soil

“…In the late '40s methods of grid points were used to model the universally large-scale atmospheric flow. During recent years ( [12], [19] and [20]), alternative methods have been used, one of these methods is the finite element method. We present here the main tools for implementations of the finite element method which belongs to the more general family of Galerkin methods.…”