# A Multigroup diffusion Solver Using Pseudo Transient Continuation for a Radiaiton-Hydrodynamic Code with Patch-Based AMR

## Abstract

We present a scheme to solve the nonlinear multigroup radiation diffusion (MGD) equations. The method is incorporated into a massively parallel, multidimensional, Eulerian radiation-hydrodynamic code with adaptive mesh refinement (AMR). The patch-based AMR algorithm refines in both space and time creating a hierarchy of levels, coarsest to finest. The physics modules are time-advanced using operator splitting. On each level, separate 'level-solve' packages advance the modules. Our multigroup level-solve adapts an implicit procedure which leads to a two-step iterative scheme that alternates between elliptic solves for each group with intra-cell group coupling. For robustness, we introduce pseudo transient continuation ({Psi}tc). We analyze the magnitude of the {Psi}tc parameter to ensure positivity of the resulting linear system, diagonal dominance and convergence of the two-step scheme. For AMR, a level defines a subdomain for refinement. For diffusive processes such as MGD, the refined level uses Dirichet boundary data at the coarse-fine interface and the data is derived from the coarse level solution. After advancing on the fine level, an additional procedure, the sync-solve (SS), is required in order to enforce conservation. The MGD SS reduces to an elliptic solve on a combined grid for a system of G equations, where G is themore »

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 940501

- Report Number(s):
- UCRL-JRNL-228646

Journal ID: ISSN 0021-9991; JCTPAH; TRN: US200824%%64

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics, vol. 227, no. 3, January 1, 2008, pp. 2154--2186; Journal Volume: 227; Journal Issue: 3

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUMM MECHANICS, GENERAL PHYSICS; AIR; ALGORITHMS; CONVERGENCE; DIFFUSION; PHYSICS; RADIATIONS; SCALARS; SIMULATION; TRANSIENTS

### Citation Formats

```
Shestakov, A I, and Offner, S R.
```*A Multigroup diffusion Solver Using Pseudo Transient Continuation for a Radiaiton-Hydrodynamic Code with Patch-Based AMR*. United States: N. p., 2007.
Web.

```
Shestakov, A I, & Offner, S R.
```*A Multigroup diffusion Solver Using Pseudo Transient Continuation for a Radiaiton-Hydrodynamic Code with Patch-Based AMR*. United States.

```
Shestakov, A I, and Offner, S R. Fri .
"A Multigroup diffusion Solver Using Pseudo Transient Continuation for a Radiaiton-Hydrodynamic Code with Patch-Based AMR". United States.
doi:. https://www.osti.gov/servlets/purl/940501.
```

```
@article{osti_940501,
```

title = {A Multigroup diffusion Solver Using Pseudo Transient Continuation for a Radiaiton-Hydrodynamic Code with Patch-Based AMR},

author = {Shestakov, A I and Offner, S R},

abstractNote = {We present a scheme to solve the nonlinear multigroup radiation diffusion (MGD) equations. The method is incorporated into a massively parallel, multidimensional, Eulerian radiation-hydrodynamic code with adaptive mesh refinement (AMR). The patch-based AMR algorithm refines in both space and time creating a hierarchy of levels, coarsest to finest. The physics modules are time-advanced using operator splitting. On each level, separate 'level-solve' packages advance the modules. Our multigroup level-solve adapts an implicit procedure which leads to a two-step iterative scheme that alternates between elliptic solves for each group with intra-cell group coupling. For robustness, we introduce pseudo transient continuation ({Psi}tc). We analyze the magnitude of the {Psi}tc parameter to ensure positivity of the resulting linear system, diagonal dominance and convergence of the two-step scheme. For AMR, a level defines a subdomain for refinement. For diffusive processes such as MGD, the refined level uses Dirichet boundary data at the coarse-fine interface and the data is derived from the coarse level solution. After advancing on the fine level, an additional procedure, the sync-solve (SS), is required in order to enforce conservation. The MGD SS reduces to an elliptic solve on a combined grid for a system of G equations, where G is the number of groups. We adapt the 'partial temperature' scheme for the SS; hence, we reuse the infrastructure developed for scalar equations. Results are presented. We consider a multigroup test problem with a known analytic solution. We demonstrate utility of {Psi}tc by running with increasingly larger timesteps. Lastly, we simulate the sudden release of energy Y inside an Al sphere (r = 15 cm) suspended in air at STP. For Y = 11 kT, we find that gray radiation diffusion and MGD produce similar results. However, if Y = 1 MT, the two packages yield different results. Our large Y simulation contradicts a long-standing theory and demonstrates the inadequacy of gray diffusion.},

doi = {},

journal = {Journal of Computational Physics, vol. 227, no. 3, January 1, 2008, pp. 2154--2186},

number = 3,

volume = 227,

place = {United States},

year = {Fri Mar 02 00:00:00 EST 2007},

month = {Fri Mar 02 00:00:00 EST 2007}

}