# zbMATH — the first resource for mathematics

On the quadrature and weak form choices in collocation type discontinuous Galerkin spectral element methods. (English) Zbl 1203.65199
Summary: We examine four nodal versions of tensor product discontinuous Galerkin spectral element approximations to systems of conservation laws for quadrilateral or hexahedral meshes. They arise from the two choices of Gauss or Gauss-Lobatto quadrature and integrate by parts once (I) or twice (II) formulations of the discontinuous Galerkin method. We show that the two formulations are in fact algebraically equivalent with either Gauss or Gauss-Lobatto quadratures when global polynomial interpolations are used to approximate the solutions and fluxes within the elements. Numerical experiments confirm the equivalence of the approximations and indicate that using Gauss quadrature with integration by parts once is the most efficient of the four approximations.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
##### Keywords:
spectral element; discontinuous Galerkin; collocation
Full Text:
##### References:
 [1] Black, K.: A conservative spectral element method for the approximation of compressible fluid flow. Kybernetika 35(1), 133–146 (1999) · Zbl 1274.76271 [2] Black, K.: Spectral element approximation of convection-diffusion type problems. Appl. Numer. Math. 33(1–4), 373–379 (2000) · Zbl 0964.65104 [3] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006) · Zbl 1093.76002 [4] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007) · Zbl 1121.76001 [5] Castel, N., Cohen, G., Durufle, M.: Application of discontinuous Galerkin spectral method on hexahedral elements for aeroacoustic. J. Comput. Acoust. 17(2), 175–196 (2009) · Zbl 1257.76047 [6] Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Springer, Berlin (1976) · Zbl 0365.76001 [7] Deng, S.: Numerical simulation of optical coupling and light propagation in coupled optical resonators with size disorder. Appl. Numer. Math. 57(5–7), 475–485 (2007) · Zbl 1190.78008 [8] Deng, S.Z., Cai, W., Astratov, V.N.: Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides. Opt. Express 12(26), 6468–6480 (2004) [9] Fagherazzi, S., Furbish, D.J., Rasetarinera, P., Hussaini, M.Y.: Application of the discontinuous spectral Galerkin method to groundwater flow. Adv. Water Resour. 27, 129–140 (2004) [10] Fagherazzi, S., Rasetarinera, P., Hussaini, M.Y., Furbish, D.J.: Numerical solution of the dam-break problem with a discontinuous Galerkin method. J. Hydraul. Eng. 130(6), 532–539 (2004) [11] Giraldo, F.X., Hesthaven, J.S., Warburton, T.: Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. J. Comput. Phys. 181(2), 499–525 (2002) · Zbl 1178.76268 [12] Giraldo, F.X., Restelli, M.: A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases. J. Comput. Phys. 227, 3849–3877 (2008) · Zbl 1194.76189 [13] Gordon, W.J., Hall, C.A.: Construction of curvilinear co-ordinate systems and their applications to mesh generation. Int. J. Numer. Methods Eng. Eng. 7, 461–477 (1973) · Zbl 0271.65062 [14] Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2008) · Zbl 1134.65068 [15] Kirby, R.M., Karniadakis, G.E.: De-aliasing on non-uniform grids: algorithms and applications. J. Comput. Phys. 191, 249–264 (2003) · Zbl 1161.76534 [16] Kopriva, D.A.: Metric identities and the discontinuous spectral element method on curvilinear meshes. J. Sci. Comput. 26(3), 301–327 (2006) · Zbl 1178.76269 [17] Kopriva, D.A., Woodruff, S.L., Hussaini, M.Y.: Discontinuous spectral element approximation of Maxwell’s Equations. In: B. Cockburn, G. Karniadakis, C.-W. Shu (eds.) Proceedings of the International Symposium on Discontinuous Galerkin Methods. Springer, New York (2000) · Zbl 0957.78023 [18] Kopriva, D.A., Woodruff, S.L., Hussaini, M.Y.: Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method. Int. J. Numer. Methods Eng. 53, 105–122 (2002) · Zbl 0994.78020 [19] Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations. Scientific Computation. Springer, Berlin (2009) · Zbl 1172.65001 [20] Lomtev, I., M Kirby, R., Karniadakis, G.E.: A discontinuous Galerkin ALE method for compressible viscous flows in moving domains. J. Comput. Phys. 155, 128–159 (1999) · Zbl 0956.76046 [21] Rasetarinera, P., Hussaini, M.Y.: An efficient implicit discontinuous spectral Galerkin method. J. Comput. Phys. 172, 718–738 (2001) · Zbl 0986.65093 [22] Rasetarinera, P., Kopriva, D.A., Hussaini, M.Y.: Discontinuous spectral element solution of acoustic radiation from thin airfoils. AIAA J. 39(11), 2070–2075 (2001) [23] Restelli, M., Giraldo, F.X.: A conservative discontinuous Galerkin semi-implicit formulation for the Navier–Stokes equations in nonhydrostatic mesoscale modeling. SIAM J. Sci. Comput. 31(3), 2231–2257 (2009) · Zbl 1405.65127 [24] Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 135(2), 250–258 (1997) · Zbl 0890.65094 [25] Stanescu, D., Farassat, F., Y Hussaini, M.: Aircraft engine noise scattering–parallel discontinuous Galerkin spectral element method. Paper 2002-0800, AIAA (2002) [26] Stanescu, D., Hussaini, M.Y., Farassat, F.: Aircraft engine noise scattering by fuselage and wings: a computational approach. J. Sound Vib. 263(2), 319–333 (2003) [27] Stanescu, D., Xu, J., Farassat, F., Hussaini, M.Y.: Computation of engine noise propagation and scattering off an aircraft. Aeroacoustics 1(4), 403–420 (2002)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.