# On the Use of Finite Difference Matrix-Vector Products in Newton-Krylov Solvers for Implicit Climate Dynamics with Spectral Elements

## Abstract

Efficient solution of global climate models requires effectively handling disparate length and time scales. Implicit solution approaches allow time integration of the physical system with a time step dictated by accuracy of the processes of interest rather than by stability governed by the fastest of the time scales present. Implicit approaches, however, require the solution of nonlinear systems within each time step. Usually, a Newton s method is applied for 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 which may show a loss of accuracy 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 based shallow-watermore »

- Authors:

- Lawrence Livermore National Laboratory (LLNL)
- ORNL

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1185916

- DOE Contract Number:
- DE-AC05-00OR22725

- Resource Type:
- Conference

- Resource Relation:
- Conference: International Conference on Computational Science, Reykjavik, Iceland, 20150601, 20150603

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Newton-Krylov; climate modeling; Jacobian vector products

### Citation Formats

```
Gardner, David, Woodward, Carol S., 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.

```
Gardner, David, Woodward, Carol S., & 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.

```
Gardner, David, Woodward, Carol S., 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:.
```

```
@article{osti_1185916,
```

title = {On the Use of Finite Difference Matrix-Vector Products in Newton-Krylov Solvers for Implicit Climate Dynamics with Spectral Elements},

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

abstractNote = {Efficient solution of global climate models requires effectively handling disparate length and time scales. Implicit solution approaches allow time integration of the physical system with a time step dictated by accuracy of the processes of interest rather than by stability governed by the fastest of the time scales present. Implicit approaches, however, require the solution of nonlinear systems within each time step. Usually, a Newton s method is applied for 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 which may show a loss of accuracy 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 based shallow-water dynamical-core of the Community Atmosphere Model (CAM).},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

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

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

}