A staggered discontinuous Galerkin method for solving SN transport equation on arbitrary polygonal grids

Abstract

This paper proposes a staggered discontinuous Galerkin (SDG) method for solving the 2D SN transport equation on arbitrary polygonal mesh. The new method allows rough grids such as highly distorted quadrilateral grids and general polygonal grids. More importantly, it is numerical flux free, different from the standard discontinuous Galerkin (DG) method using upwind flux, and thus gains the advantage of not demanding a sweeping algorithm to determine the computational ordering of all elements. The sweeping process not only requires a significant amount of computation, but also encounters deadlocks due to the presence of cycles in the corresponding directed graph on deformed three-dimensional polyhedral meshes. Additionally, there are challenges with sweeping stability on curved meshes. Our method naturally avoids these problems such that it can be generalized to high-order schemes on curved meshes. Convergence of the new method is analyzed in the linear case, and we have shown that it is optimally convergent for sufficiently smooth transport solution and appropriate total cross section. Numerical results are consistent with the theoretical analysis. The tests also show that our method employing linear elements can maintain second-order accuracy on rough grids mentioned earlier, demonstrating good robustness, and be able to model material interfaces with sharp changes of the angular flux. Moreover, the asymptotic-preserving property can be observed by scaling the cross section parameter in the thick diffusive limit problem. A test employing quadratic elements is also provided, which preliminarily demonstrates the feasibility and effectiveness of our method in higher-order scenarios.