High order interpolation of magnetic fields with vector potential reconstruction for particle simulations

TL;DR

This paper introduces a high-order, divergence-free magnetic field interpolation method using vector potential reconstruction with Hermite interpolation, improving accuracy and volume preservation in particle simulations.

Contribution

It presents a novel Hermite interpolation approach for divergence-free magnetic fields that ensures high-order continuity and enhances long-term particle trajectory accuracy.

Findings

Preserves volume and continuity in magnetic field interpolation

Improves accuracy of guiding center simulations over long times

Demonstrates effectiveness in Poincaré section analysis with finite element data

Abstract

We propose a method for interpolating divergence-free continuous magnetic fields via vector potential reconstruction using Hermite interpolation, which ensures high-order continuity for applications requiring adaptive, high-order ordinary differential equation (ODE) integrators, such as the Dormand-Prince method. The method provides C(m) continuity and achieves high-order accuracy, making it particularly suited for particle trajectory integration and Poincar\'e section analysis under optimal integration order and timestep adjustments. Through numerical experiments, we demonstrate that the Hermite interpolation method preserves volume and continuity, which are critical for conserving toroidal canonical momentum and magnetic moment in guiding center simulations, especially over long-term trajectory integration. Furthermore, we analyze the impact of insufficient derivative continuity on Runge-Kutta schemes and show how it degrades accuracy at low error tolerances, introducing discontinuity-induced truncation errors. Finally, we demonstrate performant Poincar\'e section analysis in two relevant settings of field data collocated from finite element meshes