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Impact of mass lumping on gravity and Rossby waves in 2D finite element shallow-water models. (English) Zbl 1156.76035
Summary: The goal is to evaluate the effect of mass lumping on dispersion properties of four finite element 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 consistent schemes (without lumping) and the continuous case. The $$P_{0}-P_{1}, RT_{0}$$ and $$P_1^{\text{NC}}-P_{1}$$ 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 analytical results.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76B65 Rossby waves (MSC2010) 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
##### Keywords:
dispersion relation
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##### References:
 [1] Agoshkov, Finite Elements in Fluids pp 1001– (1993) [2] CareyGF (ed.). Finite Element Modeling of Environmental Problems. Wiley: U.K., 1995. [3] Danilov, A finite-element ocean model: principles and evaluation, Ocean Modelling 6 pp 125– (2004) [4] Iskandarani, A staggered spectral finite-element model for the shallow-water equations, International Journal for Numerical Methods in Fluids 20 pp 393– (1995) · Zbl 0870.76057 [5] Lynch, A wave-equation model for finite-element tidal computations, Computers and Fluids 7 pp 207– (1979) · Zbl 0421.76013 [6] Walters, A three-dimensional finite-element model for coastal and estuarine circulation, Continental Shelf Research 12 pp 83– (1992) [7] Walters, A robust, finite-element model for hydrostatic surface water flows, Communications in Numerical Methods in Engineering 14 pp 931– (1998) · Zbl 0915.76056 [8] Hinton, A note on mass lumping and related processes in the finite element method, Earthquake Engineering and Structural Dynamics 4 pp 245– (1976) [9] Fried, Finite element mass matrix lumping by numerical integration with no convergence rate loss, International Journal of Solids and Structures 11 pp 461– (1975) · Zbl 0301.65010 [10] Foreman, A two-dimensional dispersion analysis of selected methods for solving the linearized shallow-water equations, Journal of Computational Physics 56 pp 287– (1984) · Zbl 0557.76004 [11] Le Roux, Stability/dispersion analysis of the discontinuous Galerkin linearized shallow-water system, International Journal for Numerical Methods in Fluids 48 pp 325– (2005) · Zbl 1065.76142 [12] Le Roux, Dispersion relation analysis of the PNC1-P1 finite-element pair in shallow-water models, SIAM Journal on Scientific Computing 27 pp 394– (2005) · Zbl 1141.76422 [13] Le Roux, Analysis of numerically-induced oscillations in 2D finite-element shallow-water models. Part I: inertia-gravity waves, SIAM Journal on Scientific Computing 29 pp 331– (2007) · Zbl 1387.76055 [14] Le Roux, Analysis of numerically-induced oscillations in 2D finite-element shallow-water models. Part II: free planetary waves, SIAM Journal on Scientific Computing 30 (4) pp 1971– (2008) · Zbl 1191.35024 [15] Rostand, Raviart-Thomas and Brezzi-Douglas-Marini finite element approximations of the shallow-water equations, International Journal for Numerical Methods in Fluids (2007) · Zbl 1140.76022 [16] LeBlond, Waves in the Ocean (1978) [17] Gill, Atmosphere-Ocean Dynamics (1982) [18] Longuet-Higgins, Planetary waves on a rotating sphere II, Proceedings of the Royal Society of London 284 pp 40– (1965) · Zbl 0125.26701 [19] Wentzel, A generalization of quantum conditions for the purposes of wave mechanics, Zeitschrift fur Physik 38 pp 518– (1926) [20] Raviart, Mathematical Aspects of the Finite Element Methods pp 292– (1977) [21] Hua, A noise-free finite-element scheme for the two-layer shallow-water equations, Tellus 36A pp 157– (1984) [22] Bercovier, Error estimates for the finite element method solution of the Stokes problem in the primitive variables, Numerische Mathematik 33 pp 211– (1979) · Zbl 0423.65058 [23] Sigurdsson, Computational Methods in Water Resources IX, Vol. 1: Numerical Methods in Water Resources pp 291– (1992) [24] Baranger, Connection between finite volume and mixed finite element methods, Mathematical Modelling and Numerical Analysis 30 pp 445– (1996) · Zbl 0857.65116 [25] Wajsowicz, Free planetary waves in finite-difference numerical models, Journal of Physical Oceanography 16 pp 773– (1986) [26] Rostand, Kernel analysis of the discretized finite difference and finite element shallow water models, SIAM Journal on Scientific Computing (2008) · Zbl 1191.35025
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