Local Fourier Analysis of Space-Time Relaxation and Multigrid Schemes

We consider numerical methods for generalized diffusion equations that are motivated by the transport problems arising in electron beam radiation therapy planning. While Monte Carlo methods are typically used for simulations of the forward-peaked scattering behavior of electron beams, rough calculations suggest that grid-based discretizations can provide more efficient simulations if the discretizations can be made sufficiently accurate, and optimal solvers can be found for the resulting linear systems. The multigrid method for model two-dimensional transport problems presented in [5] shows the necessary optimal scaling with some dependence on the choice of scattering kernel. In order to understand this behavior, local Fourier analysis can be applied to the two-grid cycle. Using this approach, expressions for the error-propagation operators of the coarse-grid correction and relaxation steps, projected onto the fine-grid harmonic spaces, can be found. In this paper, we consider easier problems of the form of generalized diffusion problems in space-time that are analogous to model two-dimensional transport problems. We present local Fourier analysis results for these space-time model problems and compare with convergence factors of [5]. Since one of our model problems is the diffusion equation itself, we also compare to convergence factors for the diffusion equation of [23]. The results presented here show that local Fourier analysis does not offer its usual predictivity of the convergence behavior of the diffusion equation and the generalized diffusion equations until we consider unrealistically long time intervals. 1. Introduction. Beams of charged particles penetrating matter are of interest in many fields, such as cancer therapy using electrons [11, 12], electron microscopy [18], and interstellar radiation theory [10]. Typically, these beams are modeled by Monte Carlo simulations. However, a rough theoretical complexity estimate presented in [3] suggests that deterministic, grid-based methods can provide more efficient simulations if the discretizations can be made sufficiently accurate, and if optimal solvers can be found for the resulting linear systems. Depending on the type of application, different parameter choices are used to model the problem mathematically. In this paper, we focus on parameter choices motivated by the electron-beam cancer therapy dose-calculation problem introduced in [11, 12], which implies forward-peaked scattering. Mathematically, a particle beam, or a configuration of several such beams, is modeled by a boundary value problem for a Boltzmann equation [15]. Applied to electron-beam cancer therapy planning, the equation is assumed to be linear,