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

## Abstract

We obtain here 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. Finally, the usefulness of the solution for tests of an inertial confinement fusion code is demonstrated.

- Authors:

- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1489467

- Report Number(s):
- LLNL-JRNL-753609

Journal ID: ISSN 2332-4309; 940135

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational and Theoretical Transport

- Additional Journal Information:
- Journal Name: Journal of Computational and Theoretical Transport; Journal ID: ISSN 2332-4309

- Publisher:
- Taylor and Francis

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; diffusion equation; deterministic method; radiation transport

### Citation Formats

```
Gentile, Nicholas A., and Hayes, John C. A Test Problem for Codes Solving the Discretized Diffusion Equation in Cartesian Geometry Derived Via Discrete Green’s Functions. United States: N. p., 2018.
Web. doi:10.1080/23324309.2018.1497992.
```

```
Gentile, Nicholas A., & Hayes, John C. A Test Problem for Codes Solving the Discretized Diffusion Equation in Cartesian Geometry Derived Via Discrete Green’s Functions. United States. doi:10.1080/23324309.2018.1497992.
```

```
Gentile, Nicholas A., and Hayes, John C. Sun .
"A Test Problem for Codes Solving the Discretized Diffusion Equation in Cartesian Geometry Derived Via Discrete Green’s Functions". United States. doi:10.1080/23324309.2018.1497992. https://www.osti.gov/servlets/purl/1489467.
```

```
@article{osti_1489467,
```

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

author = {Gentile, Nicholas A. and Hayes, John C.},

abstractNote = {We obtain here 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. Finally, the usefulness of the solution for tests of an inertial confinement fusion code is demonstrated.},

doi = {10.1080/23324309.2018.1497992},

journal = {Journal of Computational and Theoretical Transport},

number = ,

volume = ,

place = {United States},

year = {2018},

month = {11}

}