A sparse grid discrete ordinate discontinuous Galerkin method for the radiative transfer equation

TL;DR

This paper introduces a novel sparse grid discrete ordinate discontinuous Galerkin method for efficiently solving the high-dimensional radiative transfer equation, combining angular discretization, spatial DG discretization, and sparse grid techniques.

Contribution

The paper develops a new sparse grid DG method that reduces computational complexity for the radiative transfer equation, improving efficiency over existing methods.

Findings

The method achieves better computational efficiency.

Numerical results validate convergence and effectiveness.

The approach outperforms traditional discrete ordinate DG methods.

Abstract

The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case, leading to great difficulties in numerical simulation. To tackle this bottleneck, we first use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we make the spatial discretization by means of the discontinuous Galerkin (DG) method combined with the sparse grid method. The final linear system is solved by the block Gauss-Seidal iteration method. The computational complexity and error analysis are developed in detail, which show the new method is more efficient than the original discrete ordinate DG method. A series of numerical results are performed to validate the convergence behavior and effectiveness of the proposed method.