Solving 3D Magnetohydrostatics with RBF-FD: Applications to the Solar Corona

TL;DR

This paper introduces a novel 3D magnetohydrostatic model for the solar corona using RBF-FD discretization, addressing the challenges of solving ill-posed PDEs with complex magnetic structures.

Contribution

It develops a new numerical approach combining RBF-FD with hyperbolic-elliptic PDE solving for solar coronal magnetic fields, handling ill-posedness and nonlinearity.

Findings

Successfully reconstructs nonlinear magnetic structures in the solar corona.

Demonstrates the convergence of the method for finite forcing cases.

Highlights the delicate nature of solving ill-posed PDEs in magnetohydrostatics.

Abstract

We present a novel magnetohydrostatic numerical model that solves directly for the force-balanced magnetic field in the solar corona. This model is constructed with Radial Basis Function Finite Differences (RBF-FD), specifically 3D polyharmonic splines plus polynomials, as the core discretization. This set of PDEs is particularly difficult to solve since in the limit of the forcing going to zero it becomes ill-posed with a multitude of solutions. For the forcing equal to zero there are no numerically tractable solutions. For finite forcing, the ability to converge onto a physically viable solution is delicate as will be demonstrated. The static force-balance equations are of a hyperbolic nature, in that information of the magnetic field travels along characteristic surfaces, yet they require an elliptic type solver approach for a sparse overdetermined ill-conditioned system. As an example, we reconstruct a highly nonlinear analytic model designed to represent long-lived magnetic structures observed in the solar corona.