Impact of mass lumping on gravity and Rossby waves in 2D finite‐element shallow‐water models

Abstract

SUMMARYThe goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finiteelement velocity/surface-elevation pairs that are used to approximate the linear shallow-water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P 0 − P 1 , RT 0 and P NC 1 − P 1 pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results.

Analysis of Numerically Induced Oscillations in Two-Dimensional Finite-Element Shallow-Water Models Part II: Free Planetary Waves

“…Those are the P 0 − P 1 , P NC 1 − P 0 , and P NC 1 − P 1 pairs. In [17], by diagonalizing the velocity mass matrix, the effect of the so-called mass lumping procedure on inertia-gravity and Rossby waves is analyzed for several other FE pairs.…”

Energy‐ and enstrophy‐conserving schemes for the shallow‐water equations, based on mimetic finite elements

“…Mixed finite‐element methods are the analogue of staggered grids, since they use different finite‐element spaces for velocity and pressure. Many different combinations of finite‐element spaces have been examined in the ocean modelling literature (Le Roux, 2005, 2012; Le Roux et al, 2007, 2009; Danilov et al, 2008; Le Roux and Pouliot, 2008; Rostand and Le Roux, 2008; Comblen et al, 2010; Cotter and Ham, 2011) concentrated on combinations of spaces that have discrete versions of the div−curl and curl−grad identities, just like the C‐grid. In the numerical analysis literature, this is referred to as ‘finite‐element exterior calculus’ (Arnold et al .…”

Spurious inertial oscillations in shallow-water models

“…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].…”