# Parallel-in-time integration of Kinematic Dynamos

## TL;DR

This paper explores the application of parallel-in-time algorithms, specifically Parareal, to accelerate numerical simulations of kinematic dynamos, achieving significant speedups beyond traditional spatial parallelization methods.

## Contribution

It demonstrates the feasibility and effectiveness of using the Parareal parallel-in-time method for dynamo simulations, a novel approach in this context.

## Key findings

- Speed ups beyond spatial parallelization were achieved.
- Parareal efficiency was close to 0.3 for Galloway-Proctor flow.
- Parallel in space and time speed ups of ~300 with 1600 cores.

## Abstract

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 paper investigates the feasibility of using the parallel-in-time algorithm Parareal to speed up initial value problem simulations of the kinematic dynamo, using the open source Dedalus spectral solver. Both the time independent Roberts and time dependent Galloway-Proctor 2.5D dynamos are investigated over a range of magnetic Reynolds numbers. Speed ups beyond those possible from spatial parallelization are found in both cases. Results for the Galloway-Proctor flow are promising, with Parareal efficiency found to be close to 0.3. Roberts flow results are less efficient, but Parareal still shows some speed up over spatial parallelization alone. Parallel in space and time speed ups of $\sim300$ were found for 1600 cores for the Galloway-Proctor flow, with total parallel efficiency of $\sim0.16$.