Benchmarks for infinite medium, time dependent transport problems with isotropic scattering

TL;DR

This paper provides analytical solutions for uncollided and collided scalar flux in infinite medium, time-dependent transport problems with isotropic scattering, aiding verification of numerical methods especially for challenging delta function initial conditions.

Contribution

It introduces solutions for uncollided scalar flux for various sources, facilitating high-accuracy verification of transport codes beyond existing benchmarks.

Findings

Analytic solutions for uncollided scalar flux for specific sources.

Discussion of integration challenges and workarounds for Green's functions.

Solutions can serve as source terms for verification, similar to manufactured solutions.

Abstract

The widely used AZURV1 transport benchmarks package provides a suite of solutions to isotropic scattering transport problems with a variety of initial conditions (Ganapol 2001). Most of these solutions have an initial condition that is a Dirac delta function in space; as a result these benchmarks are challenging problems to use for verification tests in computer codes. Nevertheless, approximating a delta function in simulation often leads to low orders of convergence and the inability to test the convergence of high-order numerical methods. While there are examples in the literature of integration of these solutions as Green's functions for the transport operator to produce results for more easily simulated sources, they are limited in scope and briefly explained. For a sampling of initial conditions and sources, we present solutions for the uncollided and collided scalar flux to facilitate accurate testing of source treatment in numerical solvers. The solution for the uncollided scalar flux is found in analytic form for some sources. Since integrating the Green's functions is often nontrivial, discussion of integration difficulty and workarounds to find convergent integrals is included. Additionally, our uncollided solutions can be used as source terms in verification studies, in a similar way to the method of manufactured solutions.