# Iterative linear solvers in a 2D radiation-hydrodynamics code: Methods and performance

## Abstract

Computer codes containing both hydrodynamics and radiation play a central role in simulating both astrophysical and inertial confinement fusion (ICF) phenomena. A crucial aspect of these codes is that they require an implicit solution of the radiation diffusion equations. The authors present in this paper the results of a comparison of five different linear solvers on a range of complex radiation and radiation-hydrodynamics problems. The linear solvers used are diagonally scaled conjugate gradient, GMRES with incomplete LU preconditioning, conjugate gradient with incomplete Cholesky preconditioning, multigrid, and multigrid-preconditioned conjugate gradient. These problems involve shock propagation, opacities varying over 5--6 orders of magnitude, tabular equations of state, and dynamic ALE (Arbitrary Lagrangian Eulerian) meshes. They perform a problem size scalability study by comparing linear solver performance over a wide range of problem sizes from 1,000 to 100,000 zones. The fundamental question they address in this paper is: Is it more efficient to invert the matrix in many inexpensive steps (like diagonally scaled conjugate gradient) or in fewer expensive steps (like multigrid)? In addition, what is the answer to this question as a function of problem size and is the answer problem dependent? They find that the diagonally scaled conjugate gradient method performsmore »

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab., CA (US)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 20000627

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

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 154; Journal Issue: 1; Other Information: PBD: 1 Sep 1999; Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION; 99 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ITERATIVE METHODS; TWO-DIMENSIONAL CALCULATIONS; HYDRODYNAMICS; RADIATION TRANSPORT; INERTIAL CONFINEMENT; SHOCK WAVES; EQUATIONS OF STATE; MESH GENERATION; ALGORITHMS; PLASMA SIMULATION

### Citation Formats

```
Baldwin, C., Brown, P.N., Falgout, R., Graziani, F., and Jones, J.
```*Iterative linear solvers in a 2D radiation-hydrodynamics code: Methods and performance*. United States: N. p., 1999.
Web. doi:10.1006/jcph.1999.6290.

```
Baldwin, C., Brown, P.N., Falgout, R., Graziani, F., & Jones, J.
```*Iterative linear solvers in a 2D radiation-hydrodynamics code: Methods and performance*. United States. doi:10.1006/jcph.1999.6290.

```
Baldwin, C., Brown, P.N., Falgout, R., Graziani, F., and Jones, J. Wed .
"Iterative linear solvers in a 2D radiation-hydrodynamics code: Methods and performance". United States. doi:10.1006/jcph.1999.6290.
```

```
@article{osti_20000627,
```

title = {Iterative linear solvers in a 2D radiation-hydrodynamics code: Methods and performance},

author = {Baldwin, C. and Brown, P.N. and Falgout, R. and Graziani, F. and Jones, J.},

abstractNote = {Computer codes containing both hydrodynamics and radiation play a central role in simulating both astrophysical and inertial confinement fusion (ICF) phenomena. A crucial aspect of these codes is that they require an implicit solution of the radiation diffusion equations. The authors present in this paper the results of a comparison of five different linear solvers on a range of complex radiation and radiation-hydrodynamics problems. The linear solvers used are diagonally scaled conjugate gradient, GMRES with incomplete LU preconditioning, conjugate gradient with incomplete Cholesky preconditioning, multigrid, and multigrid-preconditioned conjugate gradient. These problems involve shock propagation, opacities varying over 5--6 orders of magnitude, tabular equations of state, and dynamic ALE (Arbitrary Lagrangian Eulerian) meshes. They perform a problem size scalability study by comparing linear solver performance over a wide range of problem sizes from 1,000 to 100,000 zones. The fundamental question they address in this paper is: Is it more efficient to invert the matrix in many inexpensive steps (like diagonally scaled conjugate gradient) or in fewer expensive steps (like multigrid)? In addition, what is the answer to this question as a function of problem size and is the answer problem dependent? They find that the diagonally scaled conjugate gradient method performs poorly with the growth of problem size, increasing in both iteration count and overall CPU time with the size of the problem and also increasing for larger time steps. For all problems considered, the multigrid algorithms scale almost perfectly (i.e., the iteration count is approximately independent of problem size and problem time step). For pure radiation flow problems (i.e., no hydrodynamics), they see speedups in CPU time of factors of {approx}15--30 for the largest problems, when comparing the multigrid solvers relative to diagonal scaled conjugate gradient.},

doi = {10.1006/jcph.1999.6290},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = 1,

volume = 154,

place = {United States},

year = {1999},

month = {9}

}