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

TitleOn the Use of Finite Difference Matrix-Vector Products in Newton-Krylov Solvers for Implicit Climate Dynamics with Spectral Elements
Publication TypeJournal Article
Year of Publication2015
Journal” Procedia Computer Scie
Volume51
Pages2036-2045
Date Published06/2015
Abstract / Summary

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 (CAM).

URLhttp://www.sciencedirect.com/science/article/pii/S1877050915012764
Journal: ” Procedia Computer Scie
Year of Publication: 2015
Volume: 51
Pages: 2036-2045
Date Published: 06/2015

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 (CAM).

Citation:
Woodward, CS, DJ Gardner, and KJ Evans.  2015.  "On the Use of Finite Difference Matrix-Vector Products in Newton-Krylov Solvers for Implicit Climate Dynamics with Spectral Elements."  ” Procedia Computer Scie 51: 2036-2045.