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A scalable multidimensional fully implicit solver for Hall magnetohydrodynamics

Luis Chacon

TL;DR

This work tackles the numerical stiffness of Hall MHD by developing a scalable fully implicit solver based on Jacobian-free Newton-Krylov methods augmented with multigrid-based, physics-informed preconditioning. A vector-potential formulation enables a clean separation of stiff EMHD dynamics, which is inverted via MG, from ion MHD dynamics solved in a Schur-complement framework, with a damped-Jacobi smoother for the EMHD block. The authors establish convergence properties, verify linear dispersion relations for whistler and kinetic Alfvén waves, and demonstrate nonlinear accuracy against the GEM reconnection benchmark, achieving strong parallel scalability up to 16,384 MPI tasks in 2D. This yields an algorithm capable of taking large implicit timesteps while maintaining robustness and efficiency, enabling large-scale HMHD simulations in astrophysical, magnetospheric, and laboratory contexts.

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.

A scalable multidimensional fully implicit solver for Hall magnetohydrodynamics

TL;DR

This work tackles the numerical stiffness of Hall MHD by developing a scalable fully implicit solver based on Jacobian-free Newton-Krylov methods augmented with multigrid-based, physics-informed preconditioning. A vector-potential formulation enables a clean separation of stiff EMHD dynamics, which is inverted via MG, from ion MHD dynamics solved in a Schur-complement framework, with a damped-Jacobi smoother for the EMHD block. The authors establish convergence properties, verify linear dispersion relations for whistler and kinetic Alfvén waves, and demonstrate nonlinear accuracy against the GEM reconnection benchmark, achieving strong parallel scalability up to 16,384 MPI tasks in 2D. This yields an algorithm capable of taking large implicit timesteps while maintaining robustness and efficiency, enabling large-scale HMHD simulations in astrophysical, magnetospheric, and laboratory contexts.

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 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.
Paper Structure (17 sections, 80 equations, 4 figures, 1 table)

This paper contains 17 sections, 80 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Vector-potential formulation of the Hall MHD model
  3. Physics-based preconditioning strategy
  4. Block structure of the linearized Hall MHD model
  5. Parabolization of the Hall MHD Jacobian matrix
  6. Numerical implementation
  7. Smoothing of the vector-potential diagonal block
  8. Numerical results
  9. Linear verification: whistler wave
  10. Linear verification: kinetic Alfvén wave
  11. Nonlinear verification: GEM challenge magnetic reconnection problem
  12. Parallel and algorithmic performance
  13. Conclusions
  14. Analysis of the smoothing property of the damped-Jacobi iteration for the linearized EMHD system
  15. Strong hyperresistivity regime: $\beta\ll\alpha$
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Time histories of perturbations of $v_{y}$, $v_{z}$, $A_{y}$, $A_{z}$ for the whistler wave linear verification test.
  • Figure 2: Time histories of perturbations of $v_{x}$, $v_{y}$, $A_{z}$ for the KAW wave linear verification test.
  • Figure 3: Nonlinear comparison of the temporal evolution of the kinetic, magnetic and thermal energies from the PIXIE3D and HiFi codes for the GEM challenge (see text for setup). The reconnected flux for PIXIE3D is compared with data from Fig. 1 of Ref. gem, with excellent agreement in the slope of the curve (which determines the peak reconnection rate) vs. the GEM results.
  • Figure 4: Snapshots of $A_{z}$ (contours) and $j_{z}$ (color) at three different times ($t=20,30,40$) for the GEM simulation.