A deterministic solver for the linear Boltzmann model of a single mono-directional proton beam
Xiaojiang Zhang, Xuemin Bai, Min Tang
TL;DR
This work develops a deterministic solver for the six-dimensional linear Boltzmann transport equation governing a single mono-directional proton beam. By decomposing protons into primary and scattering components and treating depth as a time-like variable, the authors apply Strang depth splitting into three subproblems (spatial transport, velocity diffusion, and energy slowing-down) and solve each with second-order Crank-Nicolson, enabling parallelizable computations. The energy discretization uses a two-node discontinuous Galerkin scheme within $G$ energy groups, while angular/discrete velocity space is handled via Fourier-based methods and finite-volume schemes. Numerical tests show second-order convergence in depth and energy, with IDD, spot, and LD results in close agreement with FLUKA across homogeneous and heterogeneous materials, highlighting the method’s potential for efficient, noise-free proton-dose computations in clinical settings.
Abstract
The linear Boltzmann model for proton beams is a six-dimensional partial differential equation (PDE). We propose a deterministic solver for the linear Boltzmann model based on scattering decomposition and depth-splitting methods. The main idea is to first divide the protons into primary protons and scattering protons, whose equations are derived using the source iteration method. We then treat depth as the time variable in classical time-evolutionary problems and apply the depth-splitting method. In the depth-splitting method, the full operator is decomposed into three parts, with each subsystem being easily parallelizable, which is crucial for efficient simulations. The resulting discretization exhibits second-order convergence in both the depth and energy variables. The dose distributions obtained from our solver are compared with those from Monte Carlo simulations for various materials and heterogeneous cases.