# Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion

## Abstract

An algorithm is presented for the solution of the time dependent reaction-diffusion systems which arise in non-equilibrium radiation diffusion applications. This system of nonlinear equations is solved by coupling three numerical methods, Jacobian-free Newton-Krylov, operator splitting, and multigrid linear solvers. An inexact Newton's method is used to solve the system of nonlinear equations. Since building the Jacobian matrix for problems of interest can be challenging, the authors employ a Jacobian-free implementation of Newton's method, where the action of the Jacobian matrix on a vector is approximated by a first order Taylor series expansion. Preconditioned generalized minimal residual (PGMRES) is the Krylov method used to solve the linear systems that come from the iterations of Newton's method. The preconditioner in this solution method is constructed using a physics-based divide and conquer approach, often referred to as operator splitting. This solution procedure inverts the scalar elliptic systems that make up the preconditioner using simple multigrid methods. The preconditioner also addresses the strong coupling between equations with local 2 x 2 block solves. The intra-cell coupling is applied after the inter-cell coupling has already been addressed by the elliptic solves. Results are presented using this solution procedure that demonstrate its efficiency while incurringmore »

- Authors:

- Publication Date:

- Research Org.:
- Los Alamos National Lab., NM (US)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 20067701

- DOE Contract Number:
- W-7405-ENG-36

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 160; Journal Issue: 2; Other Information: PBD: 20 May 2000; Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ALGORITHMS; NEWTON METHOD; NONLINEAR PROBLEMS; RADIATION TRANSPORT

### Citation Formats

```
Mousseau, V.A., Knoll, D.A., and Rider, W.J.
```*Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion*. United States: N. p., 2000.
Web. doi:10.1006/jcph.2000.6488.

```
Mousseau, V.A., Knoll, D.A., & Rider, W.J.
```*Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion*. United States. doi:10.1006/jcph.2000.6488.

```
Mousseau, V.A., Knoll, D.A., and Rider, W.J. Sat .
"Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion". United States. doi:10.1006/jcph.2000.6488.
```

```
@article{osti_20067701,
```

title = {Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion},

author = {Mousseau, V.A. and Knoll, D.A. and Rider, W.J.},

abstractNote = {An algorithm is presented for the solution of the time dependent reaction-diffusion systems which arise in non-equilibrium radiation diffusion applications. This system of nonlinear equations is solved by coupling three numerical methods, Jacobian-free Newton-Krylov, operator splitting, and multigrid linear solvers. An inexact Newton's method is used to solve the system of nonlinear equations. Since building the Jacobian matrix for problems of interest can be challenging, the authors employ a Jacobian-free implementation of Newton's method, where the action of the Jacobian matrix on a vector is approximated by a first order Taylor series expansion. Preconditioned generalized minimal residual (PGMRES) is the Krylov method used to solve the linear systems that come from the iterations of Newton's method. The preconditioner in this solution method is constructed using a physics-based divide and conquer approach, often referred to as operator splitting. This solution procedure inverts the scalar elliptic systems that make up the preconditioner using simple multigrid methods. The preconditioner also addresses the strong coupling between equations with local 2 x 2 block solves. The intra-cell coupling is applied after the inter-cell coupling has already been addressed by the elliptic solves. Results are presented using this solution procedure that demonstrate its efficiency while incurring minimal memory requirements.},

doi = {10.1006/jcph.2000.6488},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = 2,

volume = 160,

place = {United States},

year = {2000},

month = {5}

}