Efficient and Accurate Approximation of Inductionless Magnetohydrodynamic Flows in Fusion Problems

Abstract

We describe the numerical approximation of the incompressible inductionless magnetohydrodynamic (MHD) equations for the simulation of flow problems in fusion technologies. MHD is amultiphysics problem which consists of the coupling of the incompressible Navier-Stokes and a simplified form of the Maxwell equations through the Lorentz force. The traditional approach to solve thisproblem is to reduce the set of equations using divergence constraints to obtain Poisson equations forthe pressure and the electric potential in combination with operator splitting techniques to advance intime. Whereas this permits to reduce the complexity of each time step, a severe restriction is introducedwhich results in a scheme that is not efficient for steady problems typical of fusion technologies. Thisapproximation also makes difficult to exactly satisfy charge conservation at the discrete level, which pollutes the numerical solution, sometimes corrected partially with divergence cleaning techniques. Apartfrom discussing this issue we describe a monolithic formulation where all variables are approximated inthe appropriate function spaces, which permits to prove exact charge conservation at the discrete level.This is an important feature from a physical point of view and we present some illustrative examples.However, the monolithic formulation results in a large algebraic system whose solution is a challenge.We address this challenge using an iterative Krylov procedure with an algebraic block preconditionerin which uncoupled problems for each variable are solved. The resulting algorithm is implemented inGridapMHD.jl a parallel code written in the Julia programming language which exploits Gridap.jl, anopen source software eco-system that permits the numerical solution of partial differential equation. Wealso present a validation comparing computational with experiment results and a some performance testsillustrating the efficiency of the formulation, specially parallel scalability.