Efficient multigrid preconditioners for atmospheric flow simulations at high aspect ratio

Andreas Dedner, Eike Mueller, Robert Scheichl

Research output: Contribution to journalArticle

3 Citations (Scopus)
82 Downloads (Pure)

Abstract

Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in ‘flat’ domains. For example, in numerical weather and climate prediction, an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods, which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by Börm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219–234]. We extend the analysis to three dimensions under slightly weakened assumptions and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this, we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one-dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment, which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20 480 cores is demonstrated on the HECToR supercomputer
Original languageEnglish
Pages (from-to)76-102
Number of pages27
JournalInternational Journal for Numerical Methods in Fluids
Volume80
Issue number1
Early online date2 Jul 2015
DOIs
Publication statusPublished - 10 Jan 2016

Fingerprint

Flow simulation
Flow Simulation
Aspect Ratio
Preconditioner
Aspect ratio
Elliptic Partial Differential Equations
Tensor Product
Partial differential equations
Tensors
Pressure Correction
Grid
Vertical
Semi-Lagrangian
Spherical Shell
Thin Shells
Semi-implicit
Supercomputers
Supercomputer
Time Stepping
Efficient Solution

Keywords

  • high aspect ratio flow
  • multigrid
  • elliptic PDEs
  • atmospheric modelling
  • parallelisation
  • convergence analysis

Cite this

@article{1e3695bd125146d6b1c10807bd876a30,
title = "Efficient multigrid preconditioners for atmospheric flow simulations at high aspect ratio",
abstract = "Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in ‘flat’ domains. For example, in numerical weather and climate prediction, an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods, which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by B{\"o}rm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219–234]. We extend the analysis to three dimensions under slightly weakened assumptions and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this, we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one-dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment, which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20 480 cores is demonstrated on the HECToR supercomputer",
keywords = "high aspect ratio flow, multigrid, elliptic PDEs, atmospheric modelling, parallelisation, convergence analysis",
author = "Andreas Dedner and Eike Mueller and Robert Scheichl",
year = "2016",
month = "1",
day = "10",
doi = "10.1002/fld.4072",
language = "English",
volume = "80",
pages = "76--102",
journal = "International Journal for Numerical Methods in Fluids",
issn = "0271-2091",
publisher = "John Wiley and Sons Inc.",
number = "1",

}

TY - JOUR

T1 - Efficient multigrid preconditioners for atmospheric flow simulations at high aspect ratio

AU - Dedner, Andreas

AU - Mueller, Eike

AU - Scheichl, Robert

PY - 2016/1/10

Y1 - 2016/1/10

N2 - Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in ‘flat’ domains. For example, in numerical weather and climate prediction, an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods, which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by Börm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219–234]. We extend the analysis to three dimensions under slightly weakened assumptions and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this, we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one-dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment, which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20 480 cores is demonstrated on the HECToR supercomputer

AB - Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in ‘flat’ domains. For example, in numerical weather and climate prediction, an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods, which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by Börm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219–234]. We extend the analysis to three dimensions under slightly weakened assumptions and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this, we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one-dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment, which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20 480 cores is demonstrated on the HECToR supercomputer

KW - high aspect ratio flow

KW - multigrid

KW - elliptic PDEs

KW - atmospheric modelling

KW - parallelisation

KW - convergence analysis

U2 - 10.1002/fld.4072

DO - 10.1002/fld.4072

M3 - Article

VL - 80

SP - 76

EP - 102

JO - International Journal for Numerical Methods in Fluids

JF - International Journal for Numerical Methods in Fluids

SN - 0271-2091

IS - 1

ER -