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

TL;DR

This work adversities the challenge of maintaining a divergence-free magnetic field while enabling high-order particle simulations. It develops a Hermite-interpolation-based vector potential reconstruction that yields $C(m)$ continuity and high-order accuracy, enabling reliable use of adaptive ODE solvers like the Dormand–Prince Runge–Kutta method. The approach combines 1D Hermite theory with higher-dimensional tensor-product extensions and a rational approximation framework to reconstruct a globally solenoidal $\mathbf{B}$ from discrete data, validated through convergence tests and guiding center conservation analyses across analytic and finite-element field data. The results demonstrate improved accuracy in field evaluation, robust Poincaré section analysis, and preserved physical invariants in long-term particle trajectories, suggesting broad applicability to complex magnetic-field data in fusion plasma simulations.

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 Poincaré 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 Poincaré section analysis in two relevant settings of field data collocated from finite element meshes

Figures (10)

• Figure 1: Convergence rates for 4th order (blue curves) and 5th order (red curves) solutions to the Dormand-Prince Runge-Kutta method. Curves display numerical error from integrating \ref{['eq:ode']} for a right-hand side with second derivative discontinuity of varying magnitude $\epsilon \in [10^{-4}, 10]$. The error is shown as a function of time step size, and oscillations arise from passing through time step sizes that alternatingly lead an even and odd number of total time steps.
• Figure 2: 9th order Hermite interpolation of analytically constructed magnetic field \ref{['eq:B_from_q']}, and reconstructed poloidal flux function on a coarsest mesh. The number of grid points in each direction is given by $N_R = 5$, $N_Z = 10$, and the resulting field approximation is continuous, differentiable and divergence free.
• Figure 3: Hermite interpolation convergence for analytic magnetic fields. Solid lines display the relative error arising when computing magnetic field components and poloidal flux function as a function of grid spacing used in the Hermite interpolation. Dashed lines display the expected convergence rates. $\psi$ (lower left) and $B_R$ (upper left) approximations converge at 6th, 8th and 10th order. $B_\varphi$ (not shown) does not have an error because $RB_\varphi$ is a constant and thus is exactly represented in the polynomial basis. $B_Z$ (upper right) converges at a reduced order (5th, 7th and 9th). The reason for this is that $B_Z$ is approximated by a derivative of $u$ with respect to $R$, which truncates at the $2m_R = 4,6,8$ coefficients. Further, $\nabla B$ (lower right) converges at 4,6 and 8th order.
• Figure 4: Guiding center equations conservation-convergence test. Average, relative conservation error of $p_\varphi$ and $\mu$ as a function of the timestep. The timestep is normalized by 0.5 ms, which is the simulation's final time. Lines display the conservation errors of the 5th order Runge-Kutta method from the Dormand-Prince embedded pair of solutions. Red lines display the conservation error when background fields are set to the ITER-type VDE simulation, as displayed in FIG. (\ref{['fig:fields_mfd']}). Black lines display the conservation error when background fields are set to analytic fields with radial flux surfaces, as displayed in FIG. (\ref{['fig:fields']}). Different line styles represent the number of derivatives interpolated using the Hermite method for field reconstruction.
• Figure 5: 9th order accurate Hermite interpolation of ITER magnetic fields, reconstructing the poloidal flux function and toroidal current from a mimetic finite difference discretization. The number of grid points in each direction is $N_R = 23$, $N_Z = 48$. The resulting field approximation is continuous, differentiable and divergence free.