29 research outputs found
Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio
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 (DUNE), 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.Comment: 22 pages, 6 Figures, 2 Table
A highâorder fully explicit fluxâform semiâLagrangian shallowâwater model
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106984/1/fld3887.pd
Massively parallel solvers for elliptic partial differential equations in numerical weather and climate prediction:scalability of elliptic solvers in NWP
The demand for substantial increases in the spatial resolution of global
weather- and climate- prediction models makes it necessary to use numerically
efficient and highly scalable algorithms to solve the equations of large scale
atmospheric fluid dynamics. For stability and efficiency reasons several of the
operational forecasting centres, in particular the Met Office and the ECMWF in
the UK, use semi-implicit semi-Lagrangian time stepping in the dynamical core
of the model. The additional burden with this approach is that a three
dimensional elliptic partial differential equation (PDE) for the pressure
correction has to be solved at every model time step and this often constitutes
a significant proportion of the time spent in the dynamical core. To run within
tight operational time scales the solver has to be parallelised and there seems
to be a (perceived) misconception that elliptic solvers do not scale to large
processor counts and hence implicit time stepping can not be used in very high
resolution global models. After reviewing several methods for solving the
elliptic PDE for the pressure correction and their application in atmospheric
models we demonstrate the performance and very good scalability of Krylov
subspace solvers and multigrid algorithms for a representative model equation
with more than unknowns on 65536 cores on HECToR, the UK's national
supercomputer. For this we tested and optimised solvers from two existing
numerical libraries (DUNE and hypre) and implemented both a Conjugate Gradient
solver and a geometric multigrid algorithm based on a tensor-product approach
which exploits the strong vertical anisotropy of the discretised equation. We
study both weak and strong scalability and compare the absolute solution times
for all methods; in contrast to one-level methods the multigrid solver is
robust with respect to parameter variations.Comment: 24 pages, 7 figures, 7 table
Improving the Laplace transform integration method
We consider the Laplace transform filtering integration scheme applied to the shallow water equations, and demonstrate how it can be formulated as a finite difference scheme in the time domain. In addition, we investigate a more accurate treatment of the non linear terms. The advantages of the resulting algorithms are demonstrated by means of numerical integrations.Visiting Fellowship programme of the Natural SciencesEngineering Research Council of Canad