Fast, low-memory numerical methods for radiative transfer via hp-adaptive mesh refinement

Abstract

Numerical solutions to the radiative transfer equation are typically computationally expensive. The large expense arises because the solution has a high dimensionality with NM degrees of freedom, where the N and M arise from spatial and angular degrees of freedom, respectively. Here, a numerical method is presented that aims for fast and low-memory calculations, in the sense of computational cost and memory requirements of only O(N). The method uses a discontinuous Galerkin (DG) spectral element method and hp-adaptive mesh refinement to reduce the number of spatial degrees of freedom from N to n, thereby reducing the total cost and memory to nM, with the aim of achieving nM approximately equal to N. After this reduction in memory to O(N), in order to ensure a computational cost of O(N), a suitable preconditioner is identified and utilized. Numerical examples are presented in two spatial dimensions to allow calculation of high-resolution reference solutions for comparison, while the methodology is general and applies in either two or three spatial dimensions. The numerical examples show large memory reduction ratios N/n and fast O(N) computational cost. A variety of examples is shown, including smooth spatial variations or steep gradients, and Rayleigh (isotropic) or Mie (anisotropic) scattering. The methods could enable more tractable computations for many applications, such as medical imaging and weather and climate prediction.