368 research outputs found
Parallel-in-Time Multi-Level Integration of the Shallow-Water Equations on the Rotating Sphere
The modeling of atmospheric processes in the context of weather and climate
simulations is an important and computationally expensive challenge. The
temporal integration of the underlying PDEs requires a very large number of
time steps, even when the terms accounting for the propagation of fast
atmospheric waves are treated implicitly. Therefore, the use of
parallel-in-time integration schemes to reduce the time-to-solution is of
increasing interest, particularly in the numerical weather forecasting field.
We present a multi-level parallel-in-time integration method combining the
Parallel Full Approximation Scheme in Space and Time (PFASST) with a spatial
discretization based on Spherical Harmonics (SH). The iterative algorithm
computes multiple time steps concurrently by interweaving parallel high-order
fine corrections and serial corrections performed on a coarsened problem. To do
that, we design a methodology relying on the spectral basis of the SH to
coarsen and interpolate the problem in space. The methods are evaluated on the
shallow-water equations on the sphere using a set of tests commonly used in the
atmospheric flow community. We assess the convergence of PFASST-SH upon
refinement in time. We also investigate the impact of the coarsening strategy
on the accuracy of the scheme, and specifically on its ability to capture the
high-frequency modes accumulating in the solution. Finally, we study the
computational cost of PFASST-SH to demonstrate that our scheme resolves the
main features of the solution multiple times faster than the serial schemes
Hybridizable compatible finite element discretizations for numerical weather prediction: implementation and analysis
There is a current explosion of interest in new numerical methods for atmospheric modeling. A driving force behind this is the need to be able to simulate, with high efficiency, large-scale geophysical flows on increasingly more parallel computer systems. Many current operational models, including that of the UK Met Office, depend on orthogonal meshes, such as the latitude-longitude grid. This facilitates the development of finite difference discretizations with favorable numerical properties. However, such methods suffer from the ``pole problem," which prohibits the model to make efficient use of a large number of computing processors due to excessive concentration of grid-points at the poles.
Recently developed finite element discretizations, known as ``compatible" finite elements, avoid this issue while maintaining the key numerical properties essential for accurate geophysical simulations. Moreover, these properties can be obtained on arbitrary, non-orthogonal meshes. However, the efficient solution of the resulting discrete systems depend on transforming the mixed velocity-pressure (or velocity-pressure-buoyancy) system into an elliptic problem for the pressure. This is not so straightforward within the compatible finite element framework due to inter-element coupling.
This thesis supports the proposition that systems arising from compatible finite element discretizations can be solved efficiently using a technique known as ``hybridization." Hybridization removes inter-element coupling while maintaining the desired numerical properties. This permits the construction of sparse, elliptic problems, for which fast solver algorithms are known, using localized algebra. We first introduce the technique for compatible finite element discretizations of simplified atmospheric models. We then develop a general software abstraction for the rapid implementation and composition of hybridization methods, with an emphasis on preconditioning.
Finally, we extend the technique for a new compatible method for the full, compressible atmospheric equations used in operational models.Open Acces
A high-order scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator
ArticleThis is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Numerical Analysis following peer review. The version of record IMA J Numer Anal (2015) is available online at http://imajna.oxfordjournals.org/content/early/2015/06/16/imanum.drv021The manuscript presents a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form âu/ât=Luâu/ât=Lu, where LL is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator exp(ÏL)expâĄ(ÏL) for a relatively large time-step ÏÏ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order RungeâKutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials
Multigrid preconditioners for the hybridised discontinuous Galerkin discretisation of the shallow water equations
17 USC 105 interim-entered record; under review.The article of record as published may be found at https://doi.org/10.1016/j.jcp.2020.109948Numerical climate- and weather-prediction models require the fast solution of the
equations of fluid dynamics. Discontinuous Galerkin (DG) discretisations have several
advantageous properties. They can be used for arbitrary domains and support a structured
data layout, which is particularly important on modern chip architectures. For smooth
solutions, higher order approximations can be particularly efficient since errors decrease
exponentially in the polynomial degree. Due to the wide separation of timescales in
atmospheric dynamics, semi-implicit time integrators are highly efficient, since the implicit
treatment of fast waves avoids tight constraints on the time step size, and can therefore
improve overall efficiency. However, if implicit-explicit (IMEX) integrators are used, a large
linear system of equations has to be solved in every time step. A particular problem for DG
discretisations of velocity-pressure systems is that the normal Schur-complement reduction
to an elliptic system for the pressure is not possible since the numerical fluxes introduce
artificial diffusion terms. For the shallow water equations, which form an important model
system, hybridised DG methods have been shown to overcome this issue. However, no
attention has been paid to the efficient solution of the resulting linear system of equations.
In this paper we address this issue and show that the elliptic system for the flux unknowns
can be solved efficiently by using a non-nested multigrid algorithm. The method is
implemented in the Firedrake library and we demonstrate the excellent performance of
the algorithm both for an idealised stationary flow problem in a flat domain and for nonstationary setups in spherical geometry from the well-known testsuite in Williamson et
al. (1992) [23]. In the latter case the performance of our bespoke multigrid preconditioner
(although itself not highly optimised) is comparable to that of a highly optimised direct
solver.EPSRCEPSRCEP/L015684/1UK-Fluids network (EPSRC grant EP/N032861/1
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High-order (hybridized) discontinuous Galerkin method for geophysical flows
As computational research has grown, simulation has become a standard tool in many fields of academic and industrial areas. For example, computational fluid dynamics (CFD) tools in aerospace and research facilities are widely used to evaluate the aerodynamic performance of aircraft or wings. Weather forecasts are highly dependent on numerical weather prediction (NWP) model. However, it is still difficult to simulate the complex physical phenomena of a wide range of length and time scales with modern computational resources. In this study, we develop a robust, efficient and high-order accurate numerical methods and techniques to tackle the challenges. First, we use high-order spatial discretization using (hybridized) discontinuous Galerkin (DG) methods. The DG method combines the advantages of finite volume and finite element methods. As such, it is well-suited to problems with large gradients including shocks and with complex geometries, and large-scale simulations. However, DG typically has many degrees-of-freedoms. To mitigate the expense, we use hybridized DG (HDG) method that introduces new âtrace unknownsâ on the mesh skeleton (mortar interfaces) to eliminate the local âvolume unknownsâ with static condensation procedure and reduces globally coupled system when implicit time-stepping is required. Also, since the information between the elements is exchanged through the mesh skeleton, the mortar interfaces can be used as a glue to couple multi-phase regions, e.g., solid and fluid regions, or non-matching grids, e.g., a rotating mesh and a stationary mesh. That is the HDG method provides an efficient and flexible coupling environment compared to standard DG methods. Second, we develop an HDG-DG IMEX scheme for an efficient time integrating scheme. The idea is to divide the governing equations into stiff and nonstiff parts, implicitly treat the former with HDG methods, and explicitly treat the latter with DG methods. The HDG-DG IMEX scheme facilitates high-order temporal and spatial solutions, avoiding too small a time step. Numerical results show that the HDG-DG IMEX scheme is comparable to an explicit Runge-Kutta DG scheme in terms of accuracy while allowing for much larger timestep sizes. We also numerically observe that IMEX HDG-DG scheme can be used as a tool to suppress the high-frequency modes such as acoustic waves or fast gravity waves in atmospheric or ocean models. In short, IMEX HDG-DG methods are attractive for applications in which a fast and stable solution is important while permitting inaccurate processing of the fast modes. Third, we also develop an EXPONENTIAL DG scheme for an efficient time integrators. Similar to the IMEX method, the governing equations are separated into linear and nonlinear parts, then the two parts are spatially discretized with DG methods. Next, we analytically integrate the linear term and approximate the nonlinear term with respect to time. This method accurately handles the fast wave modes in the linear operator. To efficiently evaluate a matrix exponential, we employ the cutting-edge adaptive Krylov subspace method. Finally, we develop a sliding-mesh interface by combining nonconforming treatment and the arbitrary Lagrangian-Eulerian (ALE) scheme for simulating rotating flows, which are important to estimate the characteristics of a rotating wind turbine or understanding vortical structures shown in atmospheric or astronomical phenomena. To integrate the rotating motion of the domain, we use the ALE formulation to map the governing equation to the stationary reference domain and introduce mortar interfaces between the stationary mesh and the rotating mesh. The mortar structure on the sliding interface changes dynamically as the mesh rotatesEngineering Mechanic
Characteristic Evolution and Matching
I review the development of numerical evolution codes for general relativity
based upon the characteristic initial value problem. Progress in characteristic
evolution is traced from the early stage of 1D feasibility studies to 2D
axisymmetric codes that accurately simulate the oscillations and gravitational
collapse of relativistic stars and to current 3D codes that provide pieces of a
binary black hole spacetime. Cauchy codes have now been successful at
simulating all aspects of the binary black hole problem inside an artificially
constructed outer boundary. A prime application of characteristic evolution is
to extend such simulations to null infinity where the waveform from the binary
inspiral and merger can be unambiguously computed. This has now been
accomplished by Cauchy-characteristic extraction, where data for the
characteristic evolution is supplied by Cauchy data on an extraction worldtube
inside the artificial outer boundary. The ultimate application of
characteristic evolution is to eliminate the role of this outer boundary by
constructing a global solution via Cauchy-characteristic matching. Progress in
this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note:
updated version of arXiv:gr-qc/050809
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