Multigrid preconditioners for the hybridised discontinuous Galerkin discretisation of the shallow water equations
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Authors
Gibson, Thomas H.
Müller, Eike H.
Betteridge, Jack
Graham, Ivan G.
Subjects
Atmospheric modelling
Multigrid
Elliptic PDE
Hybridised discontinuous Galerkin
Preconditioners
Multigrid
Elliptic PDE
Hybridised discontinuous Galerkin
Preconditioners
Advisors
Date of Issue
2020-10-26
Date
26 October 2020
Publisher
Elsevier
Language
Abstract
Numerical 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.
Type
Article
Description
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.2020.109948
The article of record as published may be found at https://doi.org/10.1016/j.jcp.2020.109948
Series/Report No
Department
Organization
Naval Postgraduate School (U.S.)
Identifiers
NPS Report Number
Sponsors
EPSRC
EPSRC
EPSRC
Funder
EP/L015684/1
UK-Fluids network (EPSRC grant EP/N032861/1)
UK-Fluids network (EPSRC grant EP/N032861/1)
Format
35 p.
Citation
Betteridge, Jack, et al. "Multigrid preconditioners for the hybridised discontinuous Galerkin discretisation of the shallow water equations." Journal of Computational Physics 426 (2021): 109948.
Distribution Statement
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.