# On the use of finite difference matrix-vector products in Newton-Krylov solvers for implicit climate dynamics with spectral elements

## Abstract

Efficient solutions of global climate models require effectively handling disparate length and time scales. Implicit solution approaches allow time integration of the physical system with a step size governed by accuracy of the processes of interest rather than by stability of the fastest time scales present. Implicit approaches, however, require the solution of nonlinear systems within each time step. Usually, a Newton's method is applied to solve these systems. Each iteration of the Newton's method, in turn, requires the solution of a linear model of the nonlinear system. This model employs the Jacobian of the problem-defining nonlinear residual, but this Jacobian can be costly to form. If a Krylov linear solver is used for the solution of the linear system, the action of the Jacobian matrix on a given vector is required. In the case of spectral element methods, the Jacobian is not calculated but only implemented through matrix-vector products. The matrix-vector multiply can also be approximated by a finite difference approximation which may introduce inaccuracy in the overall nonlinear solver. In this paper, we review the advantages and disadvantages of finite difference approximations of these matrix-vector products for climate dynamics within the spectral element shallow water dynamical core ofmore »

- Authors:

- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Publication Date:

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

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1201547

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Procedia Computer Science

- Additional Journal Information:
- Journal Volume: 51; Journal Issue: C; Journal ID: ISSN 1877-0509

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 54 ENVIRONMENTAL SCIENCES; matrix-vector multiply; spectral element solvers; Newton's method

### Citation Formats

```
Woodward, Carol S., Gardner, David J., and Evans, Katherine J..
```*On the use of finite difference matrix-vector products in Newton-Krylov solvers for implicit climate dynamics with spectral elements*. United States: N. p., 2015.
Web. doi:10.1016/j.procs.2015.05.468.

```
Woodward, Carol S., Gardner, David J., & Evans, Katherine J..
```*On the use of finite difference matrix-vector products in Newton-Krylov solvers for implicit climate dynamics with spectral elements*. United States. doi:10.1016/j.procs.2015.05.468.

```
Woodward, Carol S., Gardner, David J., and Evans, Katherine J.. Thu .
"On the use of finite difference matrix-vector products in Newton-Krylov solvers for implicit climate dynamics with spectral elements". United States.
doi:10.1016/j.procs.2015.05.468. https://www.osti.gov/servlets/purl/1201547.
```

```
@article{osti_1201547,
```

title = {On the use of finite difference matrix-vector products in Newton-Krylov solvers for implicit climate dynamics with spectral elements},

author = {Woodward, Carol S. and Gardner, David J. and Evans, Katherine J.},

abstractNote = {Efficient solutions of global climate models require effectively handling disparate length and time scales. Implicit solution approaches allow time integration of the physical system with a step size governed by accuracy of the processes of interest rather than by stability of the fastest time scales present. Implicit approaches, however, require the solution of nonlinear systems within each time step. Usually, a Newton's method is applied to solve these systems. Each iteration of the Newton's method, in turn, requires the solution of a linear model of the nonlinear system. This model employs the Jacobian of the problem-defining nonlinear residual, but this Jacobian can be costly to form. If a Krylov linear solver is used for the solution of the linear system, the action of the Jacobian matrix on a given vector is required. In the case of spectral element methods, the Jacobian is not calculated but only implemented through matrix-vector products. The matrix-vector multiply can also be approximated by a finite difference approximation which may introduce inaccuracy in the overall nonlinear solver. In this paper, we review the advantages and disadvantages of finite difference approximations of these matrix-vector products for climate dynamics within the spectral element shallow water dynamical core of the Community Atmosphere Model.},

doi = {10.1016/j.procs.2015.05.468},

journal = {Procedia Computer Science},

number = C,

volume = 51,

place = {United States},

year = {Thu Jan 01 00:00:00 EST 2015},

month = {Thu Jan 01 00:00:00 EST 2015}

}