A Test Problem for Codes Solving the Discretized Diffusion Equation in Cartesian Geometry Derived Via Discrete Green’s Functions

Abstract

We obtain a solution to a zone-centered discretization of the one dimensional time-dependent diffusion equation with arbitrary initial conditions and source, constant absorption and scattering opacity, and constant zone size and time step. The solution is obtained using the discrete Green’s functions of the discretized equation. The solution of the discretized equation is useful in the testing of computer codes, because the code can be expected to agree with the solution to the discrete equation, to within small errors caused by roundoff. This is in contrast to solutions of the differential equation, with which the code results only approximately agree. The usefulness of the solution for tests of an inertial confinement fusion code is demonstrated.

Keywords:

• Diffusion equation
• deterministic method
• radiation transport

Acknowledgments

The first author would like to thank Charlie Cerjan for for helpful conversations.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the both authors was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Security, L.L.C. under Contract DE-AC52-07NA27344.