A high‐order triangular discontinuous Galerkin oceanic shallow water model

Giraldo, Francis X.; Warburton, Tim

doi:10.1002/fld.1562

Cited by 71 publications

(69 citation statements)

References 41 publications

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“…Table 2 shows the expected order of convergence OðDx kþ1 Þ, see [26]. The same experimental order of convergence is obtained by spectral element methods, see [11,18]. Further for fixed grid resolutions Dx, the errors decrease significantly for increasing k. These results are very close to the convergence studies in [16,30,15].…”

Section: Steady-state Solid Body Rotationsupporting

confidence: 75%

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: Steady-state Solid Body Rotationsupporting

confidence: 75%

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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“…For the interpolation points n i we choose the nodal sets based on the electrostatics [23] and Fekete [24] points; for simplicity we shall refer to these nodal sets collectively as Fekete points. We have already described the construction of the nodal basis functions in [9,25] and, for the sake of brevity, omit this discussion here.…”

Section: Basis Functionsmentioning

confidence: 99%

“…N dx (9) where F N = F(q N ) and S N = S(q N ) with F and S given by Equations (2) and (3), respectively. Note that Equation (9) states that q N satisfies the equation on each element e for all ∈ S where S is the finite-dimensional space…”

Section: Semi-discrete Equationsmentioning

confidence: 99%

“…We understand that the naming convention of strong versus weak is somewhat unusual, but we point out that we use this convention here in order to remain consistent with the terminology in our previous work (e.g. [9,25,30,31]). It should be understood that the weak and strong forms mentioned here both represent weak formulations of the continuous problem (perhaps weak form and strong weak form are more apt).…”

Section: Semi-discrete Equationsmentioning

confidence: 99%

“…Dupont and Lin [5], Eskilsson and Sherwin [6], Remacle et al [7], and Kubatko et al [8] constructed shallow water models on triangles using a collapsed local coordinate DG method. Giraldo and Warburton [9] developed a HO DG oceanic shallow water model on unstructured adaptive triangular grids. In that paper, we showed that the model yields exponentially convergent solutions (for smooth problems).…”

Section: Introductionmentioning

confidence: 99%

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High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

Giraldo

Restelli

2010

Numerical Methods in Fluids

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SUMMARYWe extend the explicit in time high-order triangular discontinuous Galerkin (DG) method to semi-implicit (SI) and then apply the algorithm to the two-dimensional oceanic shallow water equations; we implement high-order SI time-integrators using the backward difference formulas from orders one to six. The reason for changing the time-integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high-order DG methods. Changing the timeintegration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element-type area integrals, but also the finite volume-type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI time-integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high-order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time-explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high-order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high-order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high-order (HO) time-integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low-order time discretizations. Published in

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An hp‐adaptive discontinuous Galerkin method for shallow water flows

Eskilsson¹

2010

Numerical Methods in Fluids

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We have extended Cosmos++, a multi-dimensional unstructured adaptive mesh code for solving the covariant Newtonian and general relativistic radiation magnetohydrodynamic (MHD) equations, to accommodate both discrete finite volume and arbitrarily high order finite element structures. The new finite element implementation, called CosmosDG, is based on a discontinuous Galerkin (DG) formulation, using both entropy-based artificial viscosity and slope limiting procedures for regular-ization of shocks. High order multi-stage forward Euler and strong stability preserving Runge-Kutta time integration options complement high order spatial discretization. We have also added flexibility in the code infrastructure allowing for both adaptive mesh and adaptive basis order refinement to be performed separately or simultaneously in a local (cell-by-cell) manner. We discuss in this report the DG formulation and present tests demonstrating the robustness, accuracy, and convergence of our numerical methods applied to special and general relativistic MHD, though we note an equivalent capability currently also exists in CosmosDG for Newtonian systems.

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A high‐order triangular discontinuous Galerkin oceanic shallow water model

Cited by 71 publications

References 41 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

High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

An hp‐adaptive discontinuous Galerkin method for shallow water flows

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