Bernard, P.-E.; Deleersnijder, E.; Legat, V.; Remacle, J.-F. Dispersion analysis of discontinuous Galerkin schemes applied to Poincaré, Kelvin and Rossby waves. (English) Zbl 1300.76016 J. Sci. Comput. 34, No. 1, 26-47 (2008). Summary: A technique for analyzing dispersion properties of numerical schemes is proposed. The method is able to deal with both non dispersive or dispersive waves, i.e. waves for which the phase speed varies with wavenumber. It can be applied to unstructured grids and to finite domains with or without periodic boundary conditions.We consider the discrete version L of a linear differential operator \(\lambda\). An eigenvalue analysis of L gives eigenfunctions and eigenvalues \(({\mathbf l}_{i}, \lambda_{i})\). The spatially resolved modes are found out using a standard a posteriori error estimation procedure applied to eigenmodes. Resolved eigenfunctions \({\mathbf l}_{i}\)’s are used to determine numerical wavenumbers \(k_{i}\)’s. Eigenvalues’ imaginary parts are the wave frequencies \(\omega_{i}\) and a discrete dispersion relation \(\omega_{i}= f (k_{i})\) is constructed and compared with the exact dispersion relation of the continuous operator. Real parts of eigenvalues \(\lambda_{i}\)’s allow to compute dissipation errors of the scheme for each given class of wave.The method is applied to the discontinuous Galerkin discretization of shallow water equations in a rotating framework with a variable Coriolis force. Such a model exhibits three families of dispersive waves, including the slow Rossby waves that are usually difficult to analyze. In this paper, we present dissipation and dispersion errors for Rossby, Poincaré and Kelvin waves. We exhibit the strong superconvergence of numerical wave numbers issued of discontinuous Galerkin discretizations for all families of waves. In particular, the theoretical superconvergent rates, demonstrated for a one dimensional linear transport equation, for dissipation and dispersion errors are obtained in this two dimensional model with a variable Coriolis parameter for the Kelvin and Poincaré waves. Cited in 5 Documents MSC: 76M10 Finite element methods applied to problems in fluid mechanics 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B65 Rossby waves (MSC2010) 76U05 General theory of rotating fluids Keywords:eigenvalue analysis; a posteriori error estimation; Coriolis force; superconvergence Software:SLIM PDFBibTeX XMLCite \textit{P. E. Bernard} et al., J. Sci. Comput. 34, No. 1, 26--47 (2008; Zbl 1300.76016) Full Text: DOI References: [1] Adjerid, S., Devine, K.D., Flaherty, J.E., Krivodonova, L.: A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 1097–1112 (2002) · Zbl 0998.65098 [2] Ainsworth, M.: Dispersive and dissipative behavior of high order Discontinuous Galerkin finite element methods. J. Comput. Phys. 198(1), 106–130 (2004) · Zbl 1058.65103 [3] Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible navier-stokes equations. J. Comput. Phys. 130, 267–279 (1997) · Zbl 0871.76040 [4] Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138, 251–285 (1997) · Zbl 0902.76056 [5] Beckers, J.-M., Deleersnijder, E.: Stability of a FBTCS scheme applied to the propagation of shallow-water inertia-gravity waves on various space grids. J. Comput. Phys. 108, 95–104 (1993) · Zbl 0778.76054 [6] Bernard, P.-E., Chevaugeon, N., Legat, V., Deleersnijder, E., Remacle, J.-F.: High-order h-adaptive discontinuous Galerkin methods for ocean modeling. Ocean Dyn. 57, 109–121 (2007) [7] Chevaugeon, N., Remacle, J.-F., Galler, X., Ploumans, P., Caro, S.: Efficient discontinuous Galerkin methods for solving acoustic problems. In: 11th AIAA/CEAS Aeroacoustics Conference (2005) [8] Cockburn, B., Karniadakis, G.E., Shu, C.-W.: Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2000) [9] Gavrilov, M.B., Tosic, I.A.: Propagation of the Rossby waves on two dimensional rectangular grids. Meteorol. Atmospheric Phys. 68, 119–125 (1998) [10] Gottlieb, D., Hesthaven, J.: Spectral methods for hyperbolic problems. J. Comput. Appl. Math. 128, 83–131 (2001) · Zbl 0974.65093 [11] Hanert, E., Le Roux, D.Y., Legat, V., Deleersnijder, E.: Advection schemes for unstructured grid ocean modelling. Ocean Model. 7, 39–58 (2004) [12] Hu, F., Atkins, H.: Eigensolution analysis of the discontinuous Galerkin method with nonuniform grids I. One space dimension. J. Comput. Phys. 182(2), 516–545 (2002) · Zbl 1015.65048 [13] Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. Part 2. The h–p version of the finite element method. SIAM J. Numer. Anal. 34, 315–358 (1997) · Zbl 0884.65104 [14] Iskandarani, M., Haidvogel, D.B., Boyd, J.P.: A staggered spectral element model with application to the oceanic shallow water equations. Int. J. Numer. Methods Fluids 20, 393–414 (1995) · Zbl 0870.76057 [15] Longuet-Higgins, M.S.: Planetary waves on a rotating sphere II. Proc. R. Soc. Lond. 284, 40–68 (1965) · Zbl 0125.26701 [16] Majda, A.: Introduction to PDE’s and Waves for the Atmosphere and Ocean. American Mathematical Society (2003) · Zbl 1278.76004 [17] Marchandise, E., Chevaugeon, N., Remacle, J.-F.: Spatial and spectral superconvergence of discontinuous Galerkin method for hyperbolic problems. J. Comput. Appl. Math. (2006, in press) · Zbl 1138.65081 [18] Mesinger, F., Arakawa, A.: Numerical methods used in atmospheric models. Glob. Atmospheric Res. Programme (GARP) Publications Series No.17(1) (1976) [19] Pietrzak, J., Deleersnijder, E., Schroeter, J. (eds.): Ocean Model. (special issue) 10, 1–252 (2005). The Second International Workshop on Unstructured Mesh Numerical Modelling of Coastal, Shelf and Ocean Flows (Delft, the Netherlands, September 23–25, 2003) [20] Remacle, J.-F., Li, X., Shephard, M.S., Flaherty, J.E.: Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods. Int. J. Numer. Methods Eng. 62(7), 899–923 (2005) · Zbl 1078.76042 [21] Thompson, L., Pinsky, P.: Complex wavenumber Fourier analysis of the p-version finite element method. Comput. Mech. 13, 255–275 (1994) · Zbl 0789.73076 [22] Warburton, T., Karniadakis, G.E.: A discontinuous Galerkin method for the viscous MHD equations. J. Comput. Phys. 152, 608–641 (1999) · Zbl 0954.76051 [23] Wentzel, G.: A generalization of quantum conditions for the purposes of wave mechanics. Z. Phys. 38, 518 (1926) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.