An adaptive discretization of shallow‐water equations based on discontinuous Galerkin methods

Remacle, Jean‐François; Frazao, Sandra Soares; Li, Xiangrong; Shephard, Mark S.

doi:10.1002/fld.1204

Cited by 47 publications

(41 citation statements)

References 31 publications

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“…It has been only very recent (since 2000) that the DG method first appeared in geophysical fluid dynamics (GFD) applications. However, implementations of the DG method in GFD have remained primarily restricted to shallow water flow (see [44,35,2,20,10,12,37,40,34,23,24]). To date, there has been no published work on either SE or DG methods for nonhydrostatic mesoscale atmospheric applications.…”

Section: Introductionmentioning

confidence: 99%

A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

Giraldo

Restelli

2008

Journal of Computational Physics

225

333

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Section: Introductionmentioning

confidence: 99%

A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

Giraldo

Restelli

2008

Journal of Computational Physics

225

333

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“…Dupont and Lin [6], Eskilsson and Sherwin [7], Remacle et al [8], and Kubatko et al [9] constructed shallow water models on triangles using a collapsed local coordinate (modal) discontinuous Galerkin method; while all four of these works made extensive use of grid generation to solve their problems, the work of Remacle et al focused primarily on adaptivity with linear polynomials for dam-break problems. Giraldo et al [10] first used the DG method for the shallow water equations on the sphere, which was later followed by the construction of such models on triangular domains (see [11]); however, our work has focused on the nodal space approximations (Lagrange polynomials) rather than the more commonly used modal space approximations (which use Jacobi polynomials).…”

Section: Introductionmentioning

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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“…Here we give a very short overview of the derivation of the discretized Discontinuous Galerkin (DG) method with local Lax-Friedrichs (LxF) fluxes to simulate such a system (see [19], [20], [21] for more details). For each grid cell as many state vector as there are degree of freedoms (DOFs) in a single triangle are stored to the element-data.…”

Section: Applicationmentioning

confidence: 99%

Shared memory parallelization of fully-adaptive simulations using a dynamic tree-split and -join approach

Schreiber

Bungartz

Bäder

2012

2012 19th International Conference on High Performance Computing

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Abstract-In this work we present an approach for the parallelization of hyperbolic simulations on shared-memory architectures running on fully-adaptive grids.We tackle the parallelization problem with a dynamic sub-tree split-and join-approach by running computations on those split sub-trees in parallel using lightweight tasks. The traversal of subtrees created by tree-splittings is built upon an inherently cache efficient approach for solving hyperbolic PDEs on dynamically adaptive triangular grids using a Sierpiński space filling curve. Our communication scheme among sub-trees stores the exchangedata to/from adjacent sub-trees in a consecutive memory area which is further utilized for an improved run-length-encoded data exchange.To give results for a concrete scenario, we implemented a solver for the shallow water equations which demands for fullyadaptive grid refinement and coarsening after each time-step. Our results give detailed statistics about optimization of the split size, parallelization overhead and also strong scalability results for a simulation running on multi-socket Intel and AMD architectures.

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An adaptive discretization of shallow‐water equations based on discontinuous Galerkin methods

Cited by 47 publications

References 31 publications

A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

A high‐order triangular discontinuous Galerkin oceanic shallow water model

Shared memory parallelization of fully-adaptive simulations using a dynamic tree-split and -join approach

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