We spatially discretize the discrete ordinates radiation transport equation using high-order discontinuous Galerkin finite elements in R-Z geometry. Previous research has demonstrated first-order methods have 2nd-order spatial convergence rates in R-Z geometry. Presently, we demonstrate that higher-order (HO) methods preserve the p + 1 convergence rates on smooth solutions, where p is the finite element order. Further, we extend the use of HO finite element methods to utilize meshes with curved surfaces. We also demonstrate that meshes with curved surfaces do not degrade the observed spatial convergence rates. Finally, we exercise the methodology on a highly diffusive and scattering problem with alternating incident boundaries to show that both HO methods and mesh refinement reduce the negative scalar fluxes that result from oscillations.

我们使用RZ几何中的高阶不连续Galerkin有限元在空间上离散离散坐标辐射传输方程。先前的研究表明，一阶方法在RZ几何中具有二阶空间收敛速度。目前，我们证明了高阶（HO）方法 在光滑解上保持了p +1的收敛速度，其中p是有限元顺序。此外，我们扩展了HO有限元方法的使用，以利用具有曲面的网格。我们还证明了具有曲面的网格不会降低观察到的空间收敛速度。最后，我们在具有交替入射边界的高度扩散和散射问题上应用该方法，以表明HO方法和网格细化都减少了由振荡引起的负标量通量。