Contribution to the study of MHD convection: Theoretical framework, finite volume methods and pre-exascale simulations

Convection is a ubiquitous phenomenon in the universe, playing a key role in the structure of oceans and planetary and stellar atmospheres. This work focuses on the theoretical and numerical aspects of studying convection. First, the "all-regime" and "equilibrium" collocated methods, based on operator splitting and particularly suited for convective phenomena, are reformulated in terms of flux splitting, enhancing their flexibility. Subsequently, an extension of this method to magnetohydrodynamics is proposed. Its stability is demonstrated without relying on controlling the value of \nabla \cdot \bm{B}∇⋅B, thanks to the splitting and the use of Powell terms. The analysis for diabatic convection is extended to magnetohydrodynamics and sheared flows. A new "triple diffusive" instability and estimates of convective dynamo intensity are derived and validated through simulations using the developed methods. Finally, a large-scale simulation of convective dynamo on the supercomputer Adastra is presented. The integration of PDI and Deisa, modern I/O tools developed at Maison de la Simulation, into the HPC code "ARK" based on Kokkos+MPI is discussed. Additionally, preliminary work on the use of small neural networks to accelerate the GP-MOOD method is presented.