On the use of finite difference matrixvector products in NewtonKrylov 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 problemdefining 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 matrixvector products. The matrixvector 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 matrixvector 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 18770509
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; 54 ENVIRONMENTAL SCIENCES; matrixvector 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 matrixvector products in NewtonKrylov 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 matrixvector products in NewtonKrylov 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. 2015.
"On the use of finite difference matrixvector products in NewtonKrylov 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 matrixvector products in NewtonKrylov 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 problemdefining 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 matrixvector products. The matrixvector 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 matrixvector 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 = 2015,
month = 1
}

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 problemdefiningmore »

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