A wave propagation method for hyperbolic systems on the sphere

Rossmanith, James A.

doi:10.1016/j.jcp.2005.08.027

Cited by 60 publications

(68 citation statements)

References 58 publications

(120 reference statements)

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“…Numerous numerical methods have been proposed for next generation global atmospheric models including finite volumes [27,34], spectral elements [45,11,9,17], and DG [16,30,15] methods. We have selected the DG method for our model because it allows us to achieve high-order accuracy as in spectral elements while conserving all quantities both locally and globally as in finite volumes, see the review in [5].…”

Section: Introductionmentioning

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: Introductionmentioning

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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“…In principle, cell-vertex based FVTD methods have an advantage over cell-centered schemes as boundary conditions can be directly applied on the scatterer surface. While this is true in general, this might not be always correct for discretizations based on an 'O-O' topology [17], usually used to solve for canonical objects like sphere, almond, etc. O-O topology is normally used to discretize axisymmetric bodies in a single block structured grid framework and is described in detail in Ref.…”

Section: Scattering From 3d Bodiesmentioning

confidence: 98%

“…O-O topology is normally used to discretize axisymmetric bodies in a single block structured grid framework and is described in detail in Ref. [17]. This discretization results in degenerate points at the two poles.…”

Section: Scattering From 3d Bodiesmentioning

confidence: 99%

A Cell-Vertex Finite Volume Time Domain Method for Electromagnetic Scattering

Deore

Chatterjee²

2010

PIER M

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Abstract-A cell-vertex based finite volume scheme is used to solve the time-dependent Maxwell's equations and predict electromagnetic scattering from perfectly conducting bodies. The scheme is based on the cell-vertex finite volume integration method, originally proposed by Ni [1], for solution of the two dimensional unsteady Euler equations of gas dynamics. The resulting solution is second-order accurate in space and time, and requires cell based fluctuations to be appropriately distributed to the state vector stored at cell vertices at each time step. Results are presented for two-dimensional canonical shapes and complex three dimensional geometries. Unlike in gas dynamics, no user defined numerical damping is required in this novel cell-vertex based finite volume integration scheme when applied to the time-domain Maxwell's equations.

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“…Similar to one-dimensional case, by integrating (22), with the application of the Gauss theorem, a flux-form formulation for advancing the VIA is given as…”

Section: Extension To Two-dimensional Cartesian Gridmentioning

confidence: 99%

A multi‐moment transport model on cubed‐sphere grid

Xiao

et al. 2010

Numerical Methods in Fluids

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SUMMARYA conservative, single-cell-based semi-Lagrangian transport model is proposed in this paper. Using multimoment concept, an additional moment, i.e. volume-integrated average (VIA), is treated as the model variable besides the point value (PV) updated in the traditional semi-Lagrangian schemes. A quadratic interpolation function is constructed based on local degrees of freedom defined within each single cell. The PV moment is advanced by the semi-Lagrangian formulation, whereas the VIA moment is updated by a finite volume formulation to rigorously ensure the numerical conservation. The numerical fluxes are computed from the PV moments defined along the boundary edges of the control volume. The scheme is extended to the spherical geometry through the application of the cubed-sphere grid that eliminates the polar singularity in the conventional longitude/latitude coordinates by using the quasi-uniform grid spacing covering the whole sphere. The single-cell-based scheme is well suited for the treatment of the connections between different patches. A simple quasi-monotone limiter to the PV moment is applied to suppress non-physical oscillations. The proposed scheme has been validated via representative benchmark tests and the performance is competitive to other existing transport schemes.

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A wave propagation method for hyperbolic systems on the sphere

Cited by 60 publications

References 58 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 Cell-Vertex Finite Volume Time Domain Method for Electromagnetic Scattering

A multi‐moment transport model on cubed‐sphere grid

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