514 research outputs found
A 3D MHD model of astrophysical flows: algorithms, tests and parallelisation
In this paper we describe a numerical method designed for modelling different
kinds of astrophysical flows in three dimensions. Our method is a standard
explicit finite difference method employing the local shearing-box technique.
To model the features of astrophysical systems, which are usually
compressible, magnetised and turbulent, it is desirable to have high spatial
resolution and large domain size to model as many features as possible, on
various scales, within a particular system. In addition, the time-scales
involved are usually wide-ranging also requiring significant amounts of CPU
time.
These two limits (resolution and time-scales) enforce huge limits on
computational capabilities. The model we have developed therefore uses parallel
algorithms to increase the performance of standard serial methods. The aim of
this paper is to report the numerical methods we use and the techniques invoked
for parallelising the code. The justification of these methods is given by the
extensive tests presented herein.Comment: 17 pages with 21 GIF figures. Accepted for publication in A&
Space-time block preconditioning for incompressible flow
Parallel-in-time methods have become increasingly popular in the simulation
of time-dependent numerical PDEs, allowing for the efficient use of additional
MPI processes when spatial parallelism saturates. Most methods treat the
solution and parallelism in space and time separately. In contrast, all-at-once
methods solve the full space-time system directly, largely treating time as
simply another spatial dimension. All-at-once methods offer a number of
benefits over separate treatment of space and time, most notably significantly
increased parallelism and faster time-to-solution (when applicable). However,
the development of fast, scalable all-at-once methods has largely been limited
to time-dependent (advection-)diffusion problems. This paper introduces the
concept of space-time block preconditioning for the all-at-once solution of
incompressible flow. By extending well-known concepts of spatial block
preconditioning to the space-time setting, we develop a block preconditioner
whose application requires the solution of a space-time (advection-)diffusion
equation in the velocity block, coupled with a pressure Schur complement
approximation consisting of independent spatial solves at each time-step, and a
space-time matrix-vector multiplication. The new method is tested on four
classical models in incompressible flow. Results indicate perfect scalability
in refinement of spatial and temporal mesh spacing, perfect scalability in
nonlinear Picard iterations count when applied to a nonlinear Navier-Stokes
problem, and minimal overhead in terms of number of preconditioner applications
compared with sequential time-stepping.Comment: 28 pages, 7 figures, 4 table
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
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
Analysis of the discontinuous Galerkin method for elliptic problems on surfaces
We extend the discontinuous Galerkin (DG) framework to a linear second-order
elliptic problem on a compact smooth connected and oriented surface. An
interior penalty (IP) method is introduced on a discrete surface and we derive
a-priori error estimates by relating the latter to the original surface via the
lift introduced in Dziuk (1988). The estimates suggest that the geometric error
terms arising from the surface discretisation do not affect the overall
convergence rate of the IP method when using linear ansatz functions. This is
then verified numerically for a number of test problems. An intricate issue is
the approximation of the surface conormal required in the IP formulation,
choices of which are investigated numerically. Furthermore, we present a
generic implementation of test problems on surfaces.Comment: 21 pages, 4 figures. IMA Journal of Numerical Analysis 2013, Link to
publication: http://imajna.oxfordjournals.org/cgi/content/abstract/drs033?
ijkey=45b23qZl5oJslZQ&keytype=re
PyFR: An Open Source Framework for Solving Advection-Diffusion Type Problems on Streaming Architectures using the Flux Reconstruction Approach
High-order numerical methods for unstructured grids combine the superior
accuracy of high-order spectral or finite difference methods with the geometric
flexibility of low-order finite volume or finite element schemes. The Flux
Reconstruction (FR) approach unifies various high-order schemes for
unstructured grids within a single framework. Additionally, the FR approach
exhibits a significant degree of element locality, and is thus able to run
efficiently on modern streaming architectures, such as Graphical Processing
Units (GPUs). The aforementioned properties of FR mean it offers a promising
route to performing affordable, and hence industrially relevant,
scale-resolving simulations of hitherto intractable unsteady flows within the
vicinity of real-world engineering geometries. In this paper we present PyFR,
an open-source Python based framework for solving advection-diffusion type
problems on streaming architectures using the FR approach. The framework is
designed to solve a range of governing systems on mixed unstructured grids
containing various element types. It is also designed to target a range of
hardware platforms via use of an in-built domain specific language based on the
Mako templating engine. The current release of PyFR is able to solve the
compressible Euler and Navier-Stokes equations on grids of quadrilateral and
triangular elements in two dimensions, and hexahedral elements in three
dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented
for various benchmark flow problems, single-node performance is discussed, and
scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The
software is freely available under a 3-Clause New Style BSD license (see
www.pyfr.org)
Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows
We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable
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