Image: D.D. Holm, T. Schmah & C. Stoica. Geometric Mechanics and Symmetry:

From Finite to Infinite Dimensions, Oxford University Press (2009).

### First AGM Meeting

Geometric and Structure-Preserving Numerics

If you would like to attend the meeting on 26 April 2022 at the University of Exeter, please register by sending an email to h.alemi-ardakani@exeter.ac.uk

The meeting will be held both in-person and online.

The London Mathematical Society administers a Caring Supplementary Grant Scheme. Further information about this scheme can be found on the LMS website: www.lms.ac.uk/grants/caring-supplementary-grants

Topics expected to be discussed include:

(1) Structure-preserving numerical methods used in the Met Office models:

Modelling the atmosphere on quasi-uniform spherical meshes and geometrical underpinnings. Numerical weather prediction with compound compatible finite elements. Implementation of lateral boundary conditions in the Met Office Dynamical Core GungHo. Mimetic finite element methods for geophysical fluids.

(2) Other hydrodynamics problems: Structure-preserving schemes in ionic transport. Structure-preserving finite element integrators for compressible fluids and MHD.

26 April 2022, University of Exeter, Department of Mathematics,

Newman Building, Newman Collaborative Lecture Theatre.

The Newman Building is accessed via the Peter Chalk Centre.

The collaborative lecture theatre is in the far corner from the entrance.

Please visit the following website for directions to the Streatham Campus:

https://www.exeter.ac.uk/visit/directions/streatham/

The Newman Building is labelled Academic Building 18 on this map (in square 5F). The PDF of the map is available below:

Timetable

10:00 Registration and coffee

10:40–11:20 James Jackaman (NTNU)

11:20–12:10 Werner Bauer (Kingston University London)

12:10–13:00 Tristan Pryer (University of Bath)

13:00–14:00 Lunch

14:00–14:50 Christine Johnson (Met Office)

14:50–15:40 François Gay-Balmaz (École Normale Supérieure de Paris)

15:40–16:00 Coffee break

16:00–16:50 John Thuburn (University of Exeter)

16:50–17:40 Joshua Burby (Los Alamos National Laboratory)

Titles and Abstracts

James Jackaman (NTNU)

Preconditioned Iterative Methods for Structure-Preserving Discretisations

In recent years, there has been substantial work on the development of so-called "structure-preserving" discretisations of differential equations, where the numerical approximation reflects key conservation laws observed of the continuum solution. While such discretisations are unquestionable valuable in situations where physical relevance and fidelity are closely tied to the conservation laws, their practical utility is limited by the fact that standard iterative methods for solution of the resulting linear and nonlinear systems only resolve the underlying conserved quantities when solved to near-machine precision. In this talk, we present a generalization of the (preconditioned) flexible GMRES algorithm that can preserve arbitrarily many such conserved quantities exactly at (nearly) any stopping tolerance, with a small additional cost. Numerical results are presented for several structure-preserving finite-element discretisations of linear parabolic and hyperbolic model problems.

Werner Bauer (Kingston University London)

Towards Structure Preserving Discretizations of Stochastic Rotating Shallow Water Equations on the Sphere

We introduce a stochastic representation of the rotating shallow water equations and a corresponding structure preserving discretization. The stochastic flow model follows from using a stochastic transport principle and a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved flow components. Similarly to the deterministic case, this stochastic model (denoted as modeling under location uncertainty (LU)) conserves the global energy of any realization. Consequently, it permits us to generate an ensemble of physically relevant random simulations with a good trade-off between the representation of the model error and the ensemble’s spread. Applying a structure-preserving discretization of the deterministic part of the equations and standard finite difference/volume approximations of the stochastic terms, the resulting stochastic scheme preserves (spatially) the total energy. To address the enstrophy accumulation at the grid scale, we augment the scheme with a scale selective (energy preserving) dissipation of enstrophy, usually required to stabilize such stochastic numerical models. We compare this setup with one that applies standard biharmonic dissipation for stabilization and we study its performance for test cases of geophysical relevance.

Tristan Pryer (University of Bath)

Structure Preserving Schemes in Ionic Transport

In this work we describe the problem of ionic transport governed by the Nernst-Planck-Navier-Stokes system. We detail some of the fundamental properties of this problem, as well as the challenges involved in its approximation that manifest through numerical artefacts that are unphysical. We show that ensuring certain structures are inherited by the discretisation yields excellent approximation properties.

Christine Johnson (Met Office)

Implementation of Lateral Boundary Conditions in the Met Office Dynamical Core, GungHo

GungHo is a new dynamical core that is being developed by the Met Office to enable efficient use of future supercomputer architectures. The spatial discretization uses compatible mixed finite-elements, which allow the use of an unstructured mesh, whilst the time-discretization uses an iterated-implicit scheme, which has good stability properties. This results in solving a mixed-system of equations, which uses the associated Helmholtz equation as the preconditioner.

Lateral boundary conditions are required to enable running the dynamical core over a local region rather than the whole globe. Using simplified equations, we explore the implementation of lateral boundary conditions in GungHo and make comparisons with the current dynamical core, ENDGame, which uses a finite-difference discretization.

François Gay-Balmaz (École Normale Supérieure de Paris)

Geometric Variational Finite Element Discretization of Compressible Fluids

We review recent progress made in the development of structure preserving finite element integrators for compressible fluids and MHD. This approach combines the geometric formulation of fluid dynamics on groups of diffeomorphisms with finite element discretization techniques. A specific feature of the discrete geometric formulation is the occurrence of a nonholonomic right-invariant distribution on the discrete group of diffeomorphisms, that is shown to be isomorphic to a Raviart-Thomas finite element space. The resulting finite element discretizations correspond to weak forms of the compressible fluid equations that don't seem to have been used in the finite element literature. It extends previous work done on incompressible flows and at the lowest order on compressible fluids. We illustrate the benefits of this geometric approach and present potential future directions.

John Thuburn (University of Exeter)

Modelling the Atmosphere with Compound Compatible Finite Elements

In atmospheric models it is desirable to capture certain key properties related to conservation and balance. These physical properties are related to geometrical properties of the governing equations such as vanishing of curl of gradient, and adjointness of the divergence and negative gradient operators. Certain classes of mixed finite elements, called compatible finite elements, give rise to a discrete de Rham complex, enabling a numerical model to obtain these desirable properties.

The simplest version of this idea uses triangular mesh elements. However, the numerical wave dispersion relation on triangular grids has undesirable behavior. Wave dispersion is better on quadrilateral or hexagonal meshes. However, neither planar quadrilaterals nor planar hexagons wrap the sphere nicely; moreover, there is no simple analytical element basis function on hexagons with the required properties. Both of these issues can be avoided by constructing compound finite elements: quadrilaterals and hexagons are divided into eight and twelve triangular sub-element, respectively. The equations defining the properties of compatible finite elements are then solved on each quadrilateral or hexagonal cell using a compatible finite element discretization on the triangular sub-elements. The compound finite elements have several advantages: arbitrary polygonal elements can be constructed - we are not restricted to triangles and quadrilaterals; arbitrary polygonal elements can easily be wrapped to approximate a spherical surface; the need for 'rehabilitation' that arises with the Piola transform method is avoided.

These ideas will be illustrated using a mixed finite-element finite-volume shallow-water model that can use a hexagonal mesh or a cubed sphere mesh. The model makes use of two families of compatible finite elements, one on the primal mesh and the other on a dual mesh, each with their own discrete de Rham complex. Mapping model fields between primal and dual meshes makes use of discrete Hodge star operators that arise in a very natural way.

Joshua Burby (Los Alamos National Laboratory)

Geometric Integration of Hamiltonian Systems on Exact Symplectic Manifolds

Non-dissipative (i.e. Hamiltonian) dynamical systems freeze flux in phase space, just as highly-conductive plasma flows freeze magnetic flux. A time integrator for a non-dissipative system is symplectic when it freezes flux exactly. Symplectic integration is routine in canonical coordinates, where the flux tensor takes the simplest possible form. Much less is understood about symplectic integration in the general non-canonical case, which occurs more frequently in practice. In this talk, I will present a general approach to structure-preserving integration of noncanonical Hamiltonian systems on exact symplectic manifolds. First, the original non-canonical Hamiltonian system is embedded in a larger (essentially) canonical system as a slow manifold. Then a canonical symplectic integrator for the larger system is identified that has approximately the same slow manifold. Provided initial conditions are selected near the slow manifold, the integrator provides a good approximation of the original system. There would be a problem with this approach if the discrete-time slow manifold happened to have any normal instabilities; such instabilities would carry discrete trajectories away from the slow manifold, and the good approximation properties would break down. I will explain how this potential problem is avoided using a newly-developed theory of nearly-periodic maps. By constraining the large system's integrator to be a non-resonant nearly-periodic map, existence of a discrete-time adiabatic invariant is guaranteed. Long-time normal stability of the slow manifold then follows from a Lyapunov-type argument.

For any enquiries, please contact Hamid Alemi Ardakani.