Computation at a coordinate singularity

Prusa, Joseph M.

doi:10.1016/j.jcp.2018.01.044

Cited by 8 publications

(12 citation statements)

References 38 publications

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“…The governing Equations ( 1)-( 3) are discretised in a semi-implicit fashion, using a collocated lat-lon grid with its well-known pole problem [33]. In a nutshell, momentum equations are integrated with the trapezoidal rule…”

Section: The Discretised Modelmentioning

confidence: 99%

Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

Ackmann¹,

Dueben²,

Palmer³

et al. 2021

Preprint

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Semi-implicit grid-point models for the atmosphere and the ocean require linear solvers that are working efficiently on modern supercomputers. The huge advantage of the semi-implicit time-stepping approach is that it enables large model time-steps. This however comes at the cost of having to solve a computationally demanding linear problem each model time-step to obtain an update to the model&#8217;s pressure/fluid-thickness field. In this study, we investigate whether machine learning approaches can be used to increase the efficiency of the linear solver.Our machine learning approach aims at replacing a key component of the linear solver&#8212;the preconditioner. In the preconditioner an approximate matrix inversion is performed whose quality largely defines the linear solver&#8217;s performance. Embedding the machine-learning method within the framework of a linear solver circumvents potential robustness issues that machine learning approaches are often criticized for, as the linear solver ensures that a sufficient, pre-set level of accuracy is reached. The approach does not require prior availability of a conventional preconditioner and is highly flexible regarding complexity and machine learning design choices.Several machine learning methods of different complexity from simple linear regression to deep feed-forward neural networks are used to learn the optimal preconditioner for a shallow-water model with semi-implicit time-stepping. The shallow-water model is specifically designed to be conceptually similar to more complex atmosphere models. The machine-learning preconditioner is competitive with a conventional preconditioner and provides good results even if it is used outside of the dynamical range of the training dataset.

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Section: The Discretised Modelmentioning

confidence: 99%

Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

Ackmann¹,

Dueben²,

Palmer³

et al. 2021

Preprint

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“…The major difficulty in increasing longitudinal resolution in spherical-geometry-based GCMs is that the explicit time stepping is constrained by the clustering azimuthal cells near the pole due to the Courant-Friedrichs-Lewy (CFL) condition (Courant et al, 1928). A number of attempts have been proposed to address this coordinate singularity issue (e.g., Purser, 1988;Bouaoudia and Marcus, 1991;Williamson et al, 1992;Takacs et al, 1999;Fukagata and Kasagi, 2002;Prusa, 2018). To use a time step that is larger than the global minimum requirement from CFL conditions, one common method used in a spherical GCM is to employ a low-pass Fourier filter at polar latitudes, which removes nonphysical, high-frequency zonal waves generated due to numerical instability caused by the local violation of CFL conditions (e.g., Skamarock et al, 2008).…”

Section: Introductionmentioning

confidence: 99%

Azimuthal averaging–reconstruction filtering techniques for finite-difference general circulation models in spherical geometry

et al. 2021

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Abstract. When solving hydrodynamic equations in spherical or cylindrical geometry using explicit finite-difference schemes, a major difficulty is that the time step is greatly restricted by the clustering of azimuthal cells near the pole due to the Courant–Friedrichs–Lewy condition. This paper adapts the azimuthal averaging–reconstruction (ring average) technique to finite-difference schemes in order to mitigate the time step constraint in spherical and cylindrical coordinates. The finite-difference ring average technique averages physical quantities based on an effective grid and then reconstructs the solution back to the original grid in a piecewise, monotonic way. The algorithm is implemented in a community upper-atmospheric model, the Thermosphere–Ionosphere Electrodynamics General Circulation Model (TIEGCM), with a horizontal resolution up to 0.625∘×0.625∘ in geographic longitude–latitude coordinates, which enables the capability of resolving critical mesoscale structures within the TIEGCM. Numerical experiments have shown that the ring average technique introduces minimal artifacts in the polar region of general circulation model (GCM) solutions, which is a significant improvement compared to commonly used low-pass filtering techniques such as the fast Fourier transform method. Since the finite-difference adaption of the ring average technique is a post-solver type of algorithm, which requires no changes to the original computational grid and numerical algorithms, it has also been implemented in much more complicated models with extended physical–chemical modules such as the Coupled Magnetosphere–Ionosphere–Thermosphere (CMIT) model and the Whole Atmosphere Community Climate Model with thermosphere and ionosphere eXtension (WACCM-X). The implementation of ring average techniques in both models enables CMIT and WACCM-X to perform global simulations with a much higher resolution than that used in the community versions. The new technique is not only a significant improvement in space weather modeling capability, but it can also be adapted to more general finite-difference solvers for hyperbolic equations in spherical and polar geometries. Highlights. The ring average technique is adapted to solve the issue of clustered grid cells in polar and spherical coordinates with a finite-difference method. The ring average technique is applied to develop a 0.625∘×0.625∘ high-resolution TIEGCM and more complicated geoscientific models with polar and spherical coordinates as well as finite-difference numerical schemes. The high-resolution TIEGCM shows good capability in resolving mesoscale structures in the ionosphere–thermosphere (I–T) system.

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“…A number of studies have been carried out to investigate these structures, including the formation and evolution of polar cap patches and tongues of ionization [Basu et al, 1995;Foster et al, 2005;Zhang et al, 2013], dynamics of ionospheric irregularities [Makela and Otsuka, 2012;Sun et al, 2015], variations of polar thermospheric density anomaly [Crowley et al, 2010;Lühr et al, 2004], and the space weather effects of mesoscale electric field variability [Codrescu et al, 1995 The major difficulty in increasing longitudinal resolution in spherical geometry based GCMs is that the explicit time stepping is constrained by the clustering azimuthal cells near the pole due to the Courant-Friedrichs-Lewy (CFL) condition [Courant et al, 1928]. A number of attempts have been proposed to address this coordinate singularity issue ([e.g., Purser, 1988;Bouaoudia and Marcus, 1991;Williamson et al, 1992;Takacs, 1999;Fukagata and Kasagi, 2002;Prusa, 2018]). To use a time step that is larger than the global minimum requirement from the CFL conditions, one common method used in a spherical GCM is to employ a low-pass Fourier filter at polar latitudes, which removes non-physical, high-frequency zonal waves generated due to numerical instability caused by the local violation of the CFL conditions [e.g., Skamarock et al, 2008].…”

Section: Introductionmentioning

confidence: 99%

“…Ring Average filters used in the main The main Ring Average Algorithm in the TIE-GCM algorithms of the TIE-GCM, including the thermosphere solvers in Equations(25)(26)(27)(28)(29)(30)(31)(32)(33)(34), the ionosphere solver…”

mentioning

confidence: 99%

Development of high-resolution Thermosphere–Ionosphere Electrodynamics General Circulation Model (TIE-GCM) using Ring Average technique

Dang

Zhang

Lei

et al. 2020

Preprint

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Abstract. When solving hydrodynamic equations in spherical/cylindrical geometry using explicit finite difference schemes, a major difficulty is that the time step is greatly restricted by the clustering of azimuthal cells near the pole due to the Courant-Friedrichs-Lewy condition. This paper adapts the azimuthal averaging-reconstruction (Ring Average) technique to finite difference schemes in order to mitigate the time step constraint in spherical/cylindrical coordinates. The finite-difference Ring Average technique averages physical quantities based on an effective grid and then reconstructs the solution back to the original grid in a piece-wise, monotonic way. The algorithm is implemented in a community upper atmospheric model Thermosphere-Ionosphere Electrodynamics General Circulation Model (TIE-GCM), with horizontal resolution up to 0.625 × 0.625 in the geographic longitude-latitude coordinates, which enables the capability of resolving critical mesoscale structures within the TIE-GCM. Numerical experiments have shown that the Ring Average technique introduces minimal artifacts in the polar region of the GCM solutions, which is a significant improvement compared to the commonly used low-pass filtering techniques such as the fast Fourier transform method. Since the finite-difference adaption of the Ring Average technique is a post-solver type algorithm, which requires no changes to the original computational grid and numerical algorithms, it has also been implemented in much more complicated models with extended physical/chemical modules such as the coupled Magnetosphere Ionosphere Thermosphere (CMIT) model and the Whole Atmosphere Community Climate Model with thermosphere and ionosphere eXtension (WACCM-X). The implementation of the Ring Average techniques in both models enables CMIT and WACCM-X to perform global simulations with a much higher resolution than that used in the community versions. The new technique is not only a significant improvement in space weather modeling capability, but can also be adapted to more general finite difference solvers for hyperbolic equations in spherical/polar geometries.

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Computation at a coordinate singularity

Cited by 8 publications

References 38 publications

Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

Azimuthal averaging–reconstruction filtering techniques for finite-difference general circulation models in spherical geometry

Development of high-resolution Thermosphere–Ionosphere Electrodynamics General Circulation Model (TIE-GCM) using Ring Average technique

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