Parallel-in-time integration of astro- and geo- physical flows; application of Parareal to kinematic dynamos and Rayleigh-Bénard convection

The precise mechanisms responsible for the natural dynamos in the Earth and Sun
are still not fully understood. Numerical simulations of natural dynamos are extremely
computationally intensive, and are carried out in parameter regimes many
orders of magnitude away from real conditions. Parallelization in space is a common
strategy to speed up simulations on high performance computers, but eventually hits
a scaling limit. Additional directions of parallelization are desirable to utilise the
high number of processor cores now available. Parallel-in-time methods can deliver
speed up in addition to that offered by spatial partitioning but have not yet
been applied to dynamo simulations. This thesis investigates the feasibility of using
Parallel-in-time integration to speed up numerical simulations of dynamos.

We concentrate on applying the non-intrusive Parareal algorithm to two sub-problems
of natural dynamos: kinematic dynamos and Rayleigh-Bénard convection (RBC).
We perform real-world scaling tests on high performance computing (HPC) facilities
using the open source Dedalus spectral solver.

The kinematic dynamo problem prescribes a fluid flow and observes how the magnetic field changes over time. We investigate the time independent Roberts and
time dependent Galloway-Proctor 2.5D dynamos over a range of magnetic Reynolds
numbers. Speed ups beyond those possible from spatial parallelisation are found
in both cases. Results for the Galloway-Proctor flow are promising, with Parareal
efficiency found to be close to 0.3, while Roberts flow results are less efficient, with
efficiencies < 0.1. Parallel in space and time speed ups of 300 were found for 1600
cores for the Galloway-Proctor flow, with total parallel efficiency of 0.16.

Convective motions are thought to be the source of dynamo action in the Earth
and Sun. RBC is the archetypal problem for convection studies, and is also a
fundamental problem of fluid dynamics, with many applications to geophysical,
astrophysical, and industrial flows. We investigate Parareal for Rayleigh numbers
Ra = 10⁵, 10⁶ and 10⁷, finding limited speed up in all cases for up to ~20 processors,
whilst performance and convergence of Parareal degrades as Ra increases.

We summarise our results for the kinematic dynamos + RBC, and discuss their relevance
and implications on Parallel-in-time simulations for the full dynamo problem.