A scalable multidimensional fully implicit solver for Hall magnetohydrodynamics

TL;DR

This paper introduces a highly efficient fully implicit solver for Hall magnetohydrodynamics equations, leveraging multigrid-preconditioned Jacobian-free Newton-Krylov methods, and demonstrates its effectiveness through linear and nonlinear tests with excellent parallel scalability.

Contribution

It presents a novel, scalable implicit solver for HMHD equations using physics-informed preconditioning and vector-potential formulation, improving stability and performance.

Findings

Solver verified with wave propagation tests.

Successfully applied to GEM reconnection problem.

Achieves excellent parallel scalability up to 16384 MPI tasks.

Abstract

We propose an optimally performant fully implicit algorithm for the Hall magnetohydrodynamics (HMHD) equations based on multigrid-preconditioned Jacobian-free Newton-Krylov methods. HMHD is a challenging system to solve numerically because it supports stiff fast dispersive waves. The preconditioner is formulated using an operator-split approximate block factorization (Schur complement), informed by physics insight. We use a vector-potential formulation (instead of a magnetic field one) to allow a clean segregation of the problematic $\nabla \times \nabla \times$ operator in the electron Ohm's law subsystem. This segregation allows the formulation of an effective damped block-Jacobi smoother for multigrid. We demonstrate by analysis that our proposed block-Jacobi iteration is convergent and has the smoothing property. The resulting HMHD solver is verified linearly with wave propagation examples, and nonlinearly with the GEM challenge reconnection problem by comparison against another HMHD code. We demonstrate the excellent algorithmic and parallel performance of the algorithm up to 16384 MPI tasks in two dimensions.