The Subelement Sweeping method for radiation transport modeling on polygonal meshes

A new memory efficient way of obtaining numerical solutions to the radiation transport equation on random polygon meshes is developed and analyzed. This method is called the Subelement Sweeping method, and is applied to the discrete ordinates form of the 1-D and 2-D mono-energetic transport equations. In the Subelement Sweeping method, the coarse mesh is first subdivided into triangular subelements, and the subelement mesh is then swept to obtain subelement angular flux solutions. As the subelements are swept the scalar flux for the scattering source is interpolated from the coarse mesh. Numerical solutions on the subelements are obtained with the linear discontinuous finite element method, and the resulting angular fluxes are projected back onto the coarse mesh and accumulated into new scalar flux values. The old subelement information is then thrown away allowing the method to be memory efficient. Formulas for the interpolation from the coarse mesh fluxes to the subelement mesh fluxes and for the projection from the subelement fluxes to the coarse mesh fluxes are derived by minimizing the squared error norm between coarse mesh and subelement scalar fluxes. Asymptotic analysis is carried out in 1-D, and the Subelement Sweeping method is shown to yield a valid diffusion discretization on the coarse mesh. Asymptotic analysis is also carried out in 2-D, and the Subelement Sweeping method is shown to have the diffusion limit for orthogonal quadrilateral meshes with some simplifying assumptions. The method was implemented for the 1-D slab geometry, and numerical experiments in 1-D show that the Subelement Sweeping method is at least third order accurate and has the diffusion limit. The method was also implemented in the Capsaicin framework within the Anaheim package for 2-D polygonal meshes at Los Alamos National Laboratory. Numerical experiments in 2-D show that the Subelement Sweeping method is at least second order accurate, and suggest strongly that the diffusion limit exists for convex and non-convex polygon meshes. In conclusion, the Subelement Sweeping method obtains solutions on spatial meshes composed of composed of convex and non-convex polygon elements without using large amounts of memory storage.