Efficient elliptic solvers for higher order DG discretisations on modern architectures and applications in atmospheric modelling

  • Jack Betteridge

Student thesis: Doctoral ThesisPhD


For problems in Numerical Weather Prediction (NWP), the time to produce a solution is a critical factor.
Semi-implicit timestepping methods can speed up geophysical fluid dynamics simulations by taking larger timesteps than explicit methods.
This is possible because they treat the fast (but physically less energetic) waves implicitly, and the timestep size is not restricted by the CFL condition for these waves.
One disadvantage of this method is that an expensive linear solve must be performed at every timestep.
However, using an effective preconditioner for an iterative method significantly reduces the computational cost of this solve, making a semi-implicit scheme faster overall.

Higher order Discontinuous Galerkin (DG) methods are known for having high arithmetic intensity making them well suited for modern HPC hardware.
For smooth solutions higher order DG methods can be particularly efficient since errors decrease with a power of the polynomial degree.
However, the linear problems which arise in semi-implicit timesteppers if DG methods are used for the spatial discretisation are difficult to precondition due to the large number of coupled degrees of freedom.
This coupling arises since the numerical flux introduces off diagonal artificial diffusion terms.
Those terms would result in a dense operator if the standard Schur complement reduction to an elliptic system is used.

In this thesis we use a hybridised DG (HDG) method to eliminate the original coupling and instead couple the system of equations to a sparse operator on the trace space, which is easier to precondition.
This is achieved by considering the numerical flux variables which only lie on the facets of the mesh.
Recent work by Kang, Giraldo and Bui-Thanh [citation] solves the resulting system with a direct method.
However, this becomes impractical for high resolution simulations due to the cost of this direct
Instead, in this thesis, we solve the resulting system using a non-nested geometric multigrid technique.

In this thesis we discretise and solve the non-linear shallow water equations, an important model system in geophysical fluid dynamics, using both DG and HDG methods.
We develop a bespoke non-nested multigrid preconditioner based on work by Gopalakrishnan and Tan [citation] and implement it using Firedrake, a Python framework for solving finite element problems via code generation.
Hybridisation is performed using the Slate language, which is a part of Firedrake.
We demonstrate the effectiveness of our hybridised DG scheme with non-nested multigrid preconditioner for a range of semi-implicit IMEX timesteppers and show these provide significant improvement over traditional DG methods with the same timestepping.
Date of Award13 May 2020
Original languageEnglish
Awarding Institution
  • University of Bath
SupervisorIvan Graham (Supervisor) & Eike Mueller (Supervisor)

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