901 research outputs found
An iterative semi-implicit scheme with robust damping
An efficient, iterative semi-implicit (SI) numerical method for the time
integration of stiff wave systems is presented. Physics-based assumptions are
used to derive a convergent iterative formulation of the SI scheme which
enables the monitoring and control of the error introduced by the SI operator.
This iteration essentially turns a semi-implicit method into a fully implicit
method. Accuracy, rather than stability, determines the timestep. The scheme is
second-order accurate and shown to be equivalent to a simple preconditioning
method. We show how the diffusion operators can be handled so as to yield the
property of robust damping, i.e., dissipating the solution at all values of the
parameter \mathcal D\dt, where is a diffusion operator and \dt
the timestep. The overall scheme remains second-order accurate even if the
advection and diffusion operators do not commute. In the limit of no physical
dissipation, and for a linear test wave problem, the method is shown to be
symplectic. The method is tested on the problem of Kinetic Alfv\'en wave
mediated magnetic reconnection. A Fourier (pseudo-spectral) representation is
used. A 2-field gyrofluid model is used and an efficacious k-space SI operator
for this problem is demonstrated. CPU speed-up factors over a CFL-limited
explicit algorithm ranging from to several hundreds are obtained,
while accurately capturing the results of an explicit integration. Possible
extension of these results to a real-space (grid) discretization is discussed.Comment: Submitted to the Journal of Computational Physics. Clarifications and
caveats in response to referees, numerical demonstration of convergence rate,
generalized symplectic proo
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
Parallel-in-time integration of the shallow water equations on the rotating sphere using Parareal and MGRIT
Despite the growing interest in parallel-in-time methods as an approach to
accelerate numerical simulations in atmospheric modelling, improving their
stability and convergence remains a substantial challenge for their application
to operational models. In this work, we study the temporal parallelization of
the shallow water equations on the rotating sphere combined with time-stepping
schemes commonly used in atmospheric modelling due to their stability
properties, namely an Eulerian implicit-explicit (IMEX) method and a
semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to
investigate the performance of parallel-in-time methods, namely Parareal and
Multigrid Reduction in Time (MGRIT), when these well-established schemes are
used on the coarse discretization levels and provide insights on how they can
be improved for better performance. We begin by performing an analytical
stability study of Parareal and MGRIT applied to a linearized ordinary
differential equation depending on the choice of a coarse scheme. Next, we
perform numerical simulations of two standard tests to evaluate the stability,
convergence and speedup provided by the parallel-in-time methods compared to a
fine reference solution computed serially. We also conduct a detailed
investigation on the influence of artificial viscosity and hyperviscosity
approaches, applied on the coarse discretization levels, on the performance of
the temporal parallelization. Both the analytical stability study and the
numerical simulations indicate a poorer stability behaviour when SL-SI-SETTLS
is used on the coarse levels, compared to the IMEX scheme. With the IMEX
scheme, a better trade-off between convergence, stability and speedup compared
to serial simulations can be obtained under proper parameters and artificial
viscosity choices, opening the perspective of the potential competitiveness for
realistic models.Comment: 35 pages, 23 figure
DCMIP2016: a review of non-hydrostatic dynamical core design and intercomparison of participating models
Atmospheric dynamical cores are a fundamental component of global atmospheric modeling systems and are responsible for capturing the dynamical behavior of the Earth's atmosphere via numerical integration of the Navier-Stokes equations. These systems have existed in one form or another for over half of a century, with the earliest discretizations having now evolved into a complex ecosystem of algorithms and computational strategies. In essence, no two dynamical cores are alike, and their individual successes suggest that no perfect model exists. To better understand modern dynamical cores, this paper aims to provide a comprehensive review of 11 non-hydrostatic dynamical cores, drawn from modeling centers and groups that participated in the 2016 Dynamical Core Model Intercomparison Project (DCMIP) workshop and summer school. This review includes a choice of model grid, variable placement, vertical coordinate, prognostic equations, temporal discretization, and the diffusion, stabilization, filters, and fixers employed by each syste
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