High-Order Entropy-Based Closures for Linear Transport in Slab Geometry II: A Computational Study of the Optimization Problem

Abstract

We present a numerical algorithm to implement entropy-based ($M_N$) moment models in the context of a simple, linear kinetic equation for particles moving through a material slab. The closure for these models---as is the case for all entropy-based models---is derived through the solution of a constrained, convex optimization problem. The algorithm has two components. The first component is a discretization of the moment equations which preserves the set of realizable moments, thereby ensuring that the optimization problem has a solution (in exact arithmetic). The discretization is a second-order kinetic scheme which uses MUSCL-type limiting in space and a strong-stability-preserving, Runge--Kutta time integrator. The second component of the algorithm is a Newton-based solver for the dual optimization problem, which uses an adaptive quadrature to evaluate integrals in the dual objective and its derivatives. The accuracy of the numerical solution to the dual problem plays a key role in the time step restric...