skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

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

Journal Article · · Procedia Computer Science
 [1];  [1];  [2]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  2. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
+ Show Author Affiliations

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.

Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE
OSTI ID:
1201547
Journal Information:
Procedia Computer Science, Vol. 51, Issue C; ISSN 1877-0509
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 1 work
Citation information provided by
Web of Science

References (16)

An initial-value problem for testing numerical models of the global shallow-water equations journal January 2004
On finite difference approximation of a matrix-vector product in the Jacobian-free Newton–Krylov method journal October 2011
A Local Convergence Theory for Combined Inexact-Newton/Finite-Difference Projection Methods journal April 1987
Hybrid Krylov Methods for Nonlinear Systems of Equations journal May 1990
On Using Approximate Finite Differences in Matrix-Free Newton–Krylov Methods journal January 2008
AMIP Simulation with the CAM4 Spectral Element Dynamical Core journal February 2013
A Scalable and Adaptable Solution Framework within Components of the Community Climate System Model book January 2009
Accuracy Analysis of a Spectral Element Atmospheric Model Using a Fully Implicit Solution Framework journal August 2010
An Overlapping Schwarz Method for Spectral Element Solution of the Incompressible Navier–Stokes Equations journal May 1997
A Spectral Deferred Correction Method Applied to the Shallow Water Equations on a Sphere journal October 2013
Jacobian-free Newton–Krylov methods: a survey of approaches and applications journal January 2004
Algorithmically scalable block preconditioner for fully implicit shallow-water equations in CAM-SE journal October 2014
A Flexible Inner-Outer Preconditioned GMRES Algorithm journal March 1993
Conservative Finite-Difference Approximations of the Primitive Equations on Quasi-Uniform Spherical Grids journal February 1972
A compatible and conservative spectral element method on unstructured grids journal August 2010
A standard test set for numerical approximations to the shallow water equations in spherical geometry journal September 1992

Similar Records

Related Subjects

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