Characteristic Sweeps and Source Iteration for Charged-Particle Transport with Continuous Slowing-Down and Angular Scattering

We develop a semi-analytic deterministic framework for charged-particle transport with continuous slowing-down in energy and angular scattering. Directed transport and energy advection are treated by method-of-characteristics integration, yielding explicit directional sweeps defined by characteristic maps and inflow data. Scattering is incorporated through a fixed-point (source-iteration) scheme in which the angular gain is lagged, yielding a sequence of decoupled directional solves coupled only through angular sums. The method is formulated variationally in a transport graph space adapted to the charged particle drift. Under standard monotonicity and positivity assumptions on the stopping power and boundedness assumptions on cross sections, we establish coercivity and boundedness of the transport bilinear form, prove contraction of the source iteration under a subcriticality condition and derive a rigorous a posteriori bound for the iteration error, providing an efficient stopping criterion. We further analyse an elastic discrete-ordinates approximation, including conservation properties and a decomposition of angular error into quadrature, cone truncation and finite iteration effects. Numerical experiments for proton transport validate the characteristic sweep against an exact ballistic benchmark and demonstrate the predicted fixed-point convergence under forward-peaked scattering. Carbon-ion simulations with tabulated stopping powers and a reduced multi-species coupling illustrate Bragg peak localisation and distal tail formation driven by secondary charged fragments.

💡 Research Summary

This paper presents a semi‑analytic deterministic framework for charged‑particle transport that simultaneously accounts for continuous slowing‑down in energy and angular scattering. The authors treat the directed transport and energy advection using a method‑of‑characteristics (MoC) integration, which yields explicit directional sweeps defined by characteristic maps and inflow boundary data. Scattering is incorporated through a fixed‑point (source‑iteration) scheme in which the angular gain term is lagged; consequently each iteration consists of a collection of decoupled directional solves coupled only through angular sums.

The method is cast in a variational setting on a transport graph space adapted to the drift in space and energy. Under standard monotonicity and positivity assumptions on the stopping power S(E) (S∈W¹,∞, S>0, S′≤0) and boundedness assumptions on the total cross‑section σ_T and scattering kernel σ_S (σ_T∈L^∞, σ_S≥0), the authors prove coercivity and boundedness of the bilinear form associated with the transport operator. These properties guarantee well‑posedness of the continuous weak problem via the Lax‑Milgram theorem.

For the source iteration, a subcriticality condition is identified under which the scattering operator is a contraction in the L²‑norm. The paper provides a rigorous a‑posteriori error bound for the iteration error, ‖ψ−ψ^k‖ ≤ C‖Aψ^k−K