Prove convergence of the Anderson-accelerated outer solver

Establish convergence of the Anderson-acceleration method used to solve the nonlinear outer iteration for the coupled finite-difference Maxwell equations and particle-generated current and magnetization in the electromagnetic drift-kinetic particle-in-cell model.

Background

The numerical method advances the electromagnetic fields and drift-kinetic guiding centers through two nested nonlinear iterations. The outer iteration applies Anderson acceleration to solve for the half-step electric and magnetic fields, while each evaluation of the nonlinear residual requires advancing all guiding centers and assembling their current and magnetization contributions.

Although convergence is observed in the numerical tests reported in Section 4, the paper explicitly states that convergence of Anderson's method for this particular coupled problem has not been rigorously established. A proof or a mathematically justified convergence characterization would therefore strengthen the theoretical foundations of the solver.

References

The convergence of Anderson's method for the problem under consideration is not strictly proven, but is steadily observed in test simulations (section \ref{s4}).

— Electromagnetic drift-kinetic particle-in-cell model with energy and charge conservation for studying finite-$β$ plasmas  (2610.01429 - Morozov et al., 1 Oct 2026) in Section 3, subsection "Outer loop: implicit system of Maxwell's equations"