Monomial cubature rules since “Stroud”: a compilation

Cools, Ronald; Rabinowitz, Paul H.

doi:10.1016/0377-0427(93)90027-9

Cited by 210 publications

(152 citation statements)

References 53 publications

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“…(5) and (6) with an appropriate right hand side operator L. The evaluation of Lðq h Þ includes the evaluation of integrals over E and oE using the representation formulae (11). The integrands include both the flux function f in (1) and the surface geometry represented by Gram's determinant g. In each triangle E quadrature rules of order 2k are applied given in [41,7,28,6]. On each edge of E standard Gauss-Lobatto rules of order 2k À 1 are applied, since the electrostatic points are in fact Gauss-Lobatto points along the edges, see [22].…”

Section: Space-discrete Formulationmentioning

confidence: 99%

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

Läuter

Giraldo

Handorf

et al. 2008

Journal of Computational Physics

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a b s t r a c tA global model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R 3 , are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a twodimensional representation of tangential momentum and therefore only two discrete momentum equations.The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge-Kutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, except in a preprocessing step.The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys. 102 (1992) 211-224], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of OðDx kþ1 Þ was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integration method limits the model error. Thus, the time step Dt is restricted by both the CFL-condition and accuracy demands. Conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k.

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Section: Space-discrete Formulationmentioning

confidence: 99%

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

Läuter

Giraldo

Handorf

et al. 2008

Journal of Computational Physics

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show abstract

“…The integrands include as well the flux function f in (1) as the surface geometry represented by Gram's determinant g. In each triangle E quadrature rules of order 2k are applied given in [24], [7], [19], [6]. On each edge of E standard Gauss-Lobatto rules of order 2k − 1 are applied, since Fekete points are in fact Gauss-Lobatto points along the edges, see [2].…”

Section: Space Discrete Formulationmentioning

confidence: 99%

A Discontinuous Galerkin Method for the Shallow Water Equations in Spherical Triangular Coordinates

Läuter¹,

Giraldo²,

Dethloff³

et al. 2007

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A global barotropic model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge-Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R 3, are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional representation of tangential momentum and therefore only two discrete momentum equations.The discontinuous Galerkin method consists of an integral formulation using a Rusanov numerical flux. A strong stability-preserving third order Runge-Kutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, exept in a preprocessing step.The validation of the atmospheric model has been done considering steady-state and unsteady analytical solutions of the nonlinear shallow water equations. Experimental convergence was observed and the order of convergence k + 1 was achieved. Furthermore, the article presents a numerical experiment, for which the third order time-integration method limits the model error. Thus, the time step ∆t is restricted by both, the CFL-condition and accuracy demands. As a second step of validation, the model could reproduce a known barotropic instability caused by a small initial perturbation of a geostrophic balanced jet stream. Conservation of mass was shown up to machine precision and energy conservation converges with decreasing grid resolution and increasing polynomial order k.

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“…(8) for the optimal approximation of Debye-Wolf diffraction integral in Eq. (3) is the subject of two-dimensional cubature [17][18][19]. In this paper, we consider three different approaches to this problem.…”

Section: Fdtd Implementation Using the Tf/sf Methodsmentioning

confidence: 99%

Computation of tightly-focused laser beams in the FDTD method

Capoglu¹,

Taflove²,

Backman³

2013

Opt. Express

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Abstract:We demonstrate how a tightly-focused coherent TEM mn laser beam can be computed in the finite-difference time-domain (FDTD) method. The electromagnetic field around the focus is decomposed into a plane-wave spectrum, and approximated by a finite number of plane waves injected into the FDTD grid using the total-field/scattered-field (TF/SF) method. We provide an error analysis, and guidelines for the discrete approximation. We analyze the scattering of the beam from layered spaces and individual scatterers. The described method should be useful for the simulation of confocal microscopy and optical data storage. An implementation of the method can be found in our free and open source FDTD software ("Angora").

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Monomial cubature rules since “Stroud”: a compilation

Cited by 210 publications

References 53 publications

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

A Discontinuous Galerkin Method for the Shallow Water Equations in Spherical Triangular Coordinates

Computation of tightly-focused laser beams in the FDTD method

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