On the Use of Finite Difference MatrixVector Products in NewtonKrylov 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 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 finitedifference which may show a loss of accuracy in the overall nonlinear solver. In this paper, we review the advantages and disadvantages of finitedifference approximations of these matrixvector products for climate dynamics within the spectralelement based shallowwatermore »
 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:
 DEAC0500OR22725
 Resource Type:
 Conference
 Resource Relation:
 Conference: International Conference on Computational Science, Reykjavik, Iceland, 20150601, 20150603
 Country of Publication:
 United States
 Language:
 English
 Subject:
 NewtonKrylov; climate modeling; Jacobian vector products
Citation Formats
Gardner, David, Woodward, Carol S., 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.
Gardner, David, Woodward, Carol S., & Evans, Katherine J. On the Use of Finite Difference MatrixVector Products in NewtonKrylov Solvers for Implicit Climate Dynamics with Spectral Elements. United States.
Gardner, David, Woodward, Carol S., 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:.
@article{osti_1185916,
title = {On the Use of Finite Difference MatrixVector Products in NewtonKrylov 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 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 finitedifference which may show a loss of accuracy in the overall nonlinear solver. In this paper, we review the advantages and disadvantages of finitedifference approximations of these matrixvector products for climate dynamics within the spectralelement based shallowwater dynamicalcore of the Community Atmosphere Model (CAM).},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = 2015,
month = 1
}

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 thismore »

Adaptive sparse linear solvers for implicit CFD using NewtonKrylov algorithms.
No abstract prepared.