Comparison of Spatial Discretization Methods for Solving the S_N Equations Using a Three-Dimensional Method of Manufactured Solutions Benchmark Suite with Escalating Order of Nonsmoothness

A comparison of the accuracy and computational efficiency of spatial discretization methods of the three-dimensional S_N equations is conducted, including discontinuous Galerkin finite element methods, the arbitrarily high-order transport method of nodal type (AHOTN), the linear-linear method, the linear-nodal (LN) method, and the higher-order diamond difference method. For this purpose, we have developed a suite of method of manufactured solutions benchmarks that provides an exact solution of the S_N equations even in the presence of scattering. Most importantly, our benchmark suite permits the user to set an arbitrary level of smoothness of the exact solution across the singular characteristics. Our study focuses on the computational efficiency of the considered spatial discretization methods.

Numerical results indicate that the best-performing method depends on the norm used to measure the discretization error. We employ discrete L_p norms and integral error norms in this work. For configurations with continuous exact angular flux, high-order AHOTNs perform best under L_p error norms, while the LN method performs best when measured by integral error norms. When the angular flux is discontinuous, a new singular-characteristic tracking method for three-dimensional geometries performs best among the considered methods.