Discontinuous Least-Squares Spatial Discretization Schemes for the One-Dimensional Slab-Geometry Sn Equations

We derive three new linear-discontinuous least-squares discretizations for the Sn equations in one-dimensional slab geometry. Standard least-squares methods are not compatible with discontinuous trial spaces, and they are also generally not conservative. Our new methods are unique in that they are based upon a least-squares minimization principle, use a discontinuous trial space, are conservative, and retain the structure of standard Sn spatial discretization schemes. To our knowledge, conservative least-squares spatial discretization schemes have not previously been developed for the Sn equations. We compare our new methods both theoretically and numerically to the linear-discontinuous Galerkin method and the lumped linear-discontinuous Galerkin method. We find that one of our schemes is clearly superior to the other two and offers certain advantages over both of the Galerkin schemes.