The Lagrange-Galerkin method for the two-dimensional shallow water equations on adaptive grids

Giraldo, Francis X.

doi:10.1002/1097-0363(20000730)33:6<789::aid-fld29>3.0.co;2-1

Cited by 35 publications

(33 citation statements)

References 21 publications

(30 reference statements)

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“…Thus where the magnitude of the gradient vector is large, the mesh is refined and where it is small, the mesh is coarsened. Further information about this adaptive grid generator can be found in [42,43]. While this particular adaptive grid generator only works for linear elements there are, however, more sophisticated adaptive mesh generators which can handle high-order elements; one example is the AMATOS package developed by Behrens et al [41].…”

Section: Advantages Of Unstructured Triangulationsmentioning

confidence: 99%

A high‐order triangular discontinuous Galerkin oceanic shallow water model

Giraldo

Warburton

2007

Numerical Methods in Fluids

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SUMMARYA high-order triangular discontinuous Galerkin (DG) method is applied to the two-dimensional oceanic shallow water equations. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. Both the area and boundary integrals are evaluated using order 2N Gauss cubature rules. The use of exact integration for the area integrals leads naturally to a full mass matrix; however, by using straight-edged triangles we eliminate the mass matrix completely from the discrete equations. Besides obviating the need for a mass matrix, triangular elements offer other obvious advantages in the construction of oceanic shallow water models, specifically the ability to use unstructured grids in order to better represent the continental coastlines for use in tsunami modeling. In this paper, we focus primarily on testing the discrete spatial operators by using six test cases-three of which have analytic solutions. The three tests having analytic solutions show that the high-order triangular DG method exhibits exponential convergence. Furthermore, comparisons with a spectral element model show that the DG model is superior for all polynomial orders and test cases considered.

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Section: Advantages Of Unstructured Triangulationsmentioning

confidence: 99%

A high‐order triangular discontinuous Galerkin oceanic shallow water model

Giraldo

Warburton

2007

Numerical Methods in Fluids

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“…Dietachmayer and Droegemeier (1992) use this global grid redistribution technique to increase resolution in areas where the estimated solution error is high. Giraldo (2000) and Iselin et al (2002) have also applied this type of dynamic adaptation for the 2D shallow-water equations and advection problems, respectively. More recently, Kühnlein et al (2012) implemented r adaptivity within a 3D Cartesian framework, Bauer et al (2014) used r-refinement grids guided by error estimates in a shallow-water model, and Weller et al (2016) demonstrated r-refinement use on the sphere.…”

Section: Introductionmentioning

confidence: 99%

Analyzing the Adaptive Mesh Refinement (AMR) Characteristics of a High-Order 2D Cubed-Sphere Shallow-Water Model

Ferguson

Jablonowski

Johansen

et al. 2016

Monthly Weather Review

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Adaptive mesh refinement (AMR) is a technique that has been featured only sporadically in atmospheric science literature. This paper aims to demonstrate the utility of AMR for simulating atmospheric flows. Several test cases are implemented in a 2D shallow-water model on the sphere using the Chombo-AMR dynamical core. This high-order finite-volume model implements adaptive refinement in both space and time on a cubed-sphere grid using a mapped-multiblock mesh technique. The tests consist of the passive advection of a tracer around moving vortices, a steady-state geostrophic flow, an unsteady solid-body rotation, a gravity wave impinging on a mountain, and the interaction of binary vortices. Both static and dynamic refinements are analyzed to determine the strengths and weaknesses of AMR in both complex flows with small-scale features and large-scale smooth flows. The different test cases required different AMR criteria, such as vorticity or height-gradient based thresholds, in order to achieve the best accuracy for cost. The simulations show that the model can accurately resolve key local features without requiring global high-resolution grids. The adaptive grids are able to track features of interest reliably without inducing noise or visible distortions at the coarse-fine interfaces. Furthermore, the AMR grids keep any degradations of the large-scale smooth flows to a minimum.

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“…The dynamic adaptation capabilities were implemented to the MPDATA model by Iselin and coworkers [22]. Another example of AMR which uses triangular elements is the work of Giraldo [23] who used the Lagrange-Galerkin method. In all of these approaches the mesh is obtained by a Delaunay triangulation of the domain given some mesh size criteria.…”

Section: Amr In Atmospheric Simulationsmentioning

confidence: 99%