Poloidal-toroidal decomposition in a finite cylinder. I. Influence matrices for the magnetohydrodynamic equations

TL;DR

This paper develops a mathematical framework using poloidal-toroidal decomposition in a finite cylinder to efficiently solve magnetohydrodynamic equations, employing influence matrices to decouple complex PDE systems and match boundary conditions.

Contribution

It introduces an influence matrix approach to decouple higher-order PDEs in MHD within a finite cylinder, simplifying boundary condition handling and eliminating exterior discretization.

Findings

Successfully decouples PDE systems using influence matrices.

Matches magnetic fields to exterior vacuum via Dirichlet-to-Neumann mapping.

Achieves well-conditioned influence matrices for numerical stability.

Abstract

The Navier-Stokes equations and magnetohydrodynamics equations are written in terms of poloidal and toroidal potentials in a finite cylinder. This formulation insures that the velocity and magnetic fields are divergence-free by construction, but leads to systems of partial differential equations of higher order, whose boundary conditions are coupled. The influence matrix technique is used to transform these systems into decoupled parabolic and elliptic problems. The magnetic field in the induction equation is matched to that in an exterior vacuum by means of the Dirichlet-to-Neumann mapping, thus eliminating the need to discretize the exterior. The influence matrix is scaled in order to attain an acceptable condition number.