A non-negative moment-preserving spatial discretization scheme for the linearized Boltzmann transport equation in 1-D and 2-D Cartesian geometries

We present a new nonlinear spatial finite-element method for the linearized Boltzmann transport equation with Sn angular discretization in 1-D and 2-D Cartesian geometries. This method has two central characteristics. First, it is equivalent to the linear-discontinuous (LD) Galerkin method whenever that method yields a strictly non-negative solution. Second, it always satisfies both the zeroth and first spatial moment equations. Because it yields the LD solution when that solution is non-negative, one might interpret our method as a classical fix-up to the LD scheme. However, fix-up schemes for the LD equations derived in the past have given up solution of the first moment equations when the LD solution is negative in order to satisfy positivity in a simple manner. We present computational results comparing our method in 1-D to the strictly non-negative linear exponential-discontinuous method and to the LD method. We present computational results in 2-D comparing our method to a recently developed LD fix-up scheme and to the LD scheme. It is demonstrated that our method is a valuable alternative to existing methods.