A Multigrid Method of the Second Kind for Solving Linear Systems of Odes Discretized by Continuous Runge-Kutta Methods

2002 ◽  
Vol 79 (7) ◽  
pp. 823-839
Author(s):  
S. Maset
Keyword(s):  
Linear Systems ◽  
Multigrid Method ◽  
Systems Of Odes ◽  
Runge Kutta

On the multigrid cycle strategy with approximate inverse smoothing

2014 ◽  
Vol 31 (1) ◽  
pp. 110-122 ◽  
Author(s):  
George A. Gravvanis ◽  
Christos K. Filelis-Papadopoulos
Keyword(s):  
Finite Difference ◽  
Linear Systems ◽  
Multigrid Method ◽  
Multigrid Methods ◽  
Inverse Matrix ◽  
Sparse Linear Systems ◽  
Approximate Inverse ◽  
Content Type ◽  
Sparsity Pattern ◽  
Approximate Inverses

Purpose – The purpose of this paper is to propose multigrid methods in conjunction with explicit approximate inverses with various cycles strategies and comparison with the other smoothers. Design/methodology/approach – The main motive for the derivation of the various multigrid schemes lies in the efficiency of the multigrid methods as well as the explicit approximate inverses. The combination of the various multigrid cycles with the explicit approximate inverses as smoothers in conjunction with the dynamic over/under relaxation (DOUR) algorithm results in efficient schemes for solving large sparse linear systems derived from the discretization of partial differential equations (PDE). Findings – Application of the proposed multigrid methods on two-dimensional boundary value problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than the V-cycle multigrid schemes presented in a recent report (Filelis-Papadopoulos and Gravvanis). Research limitations/implications – The limitations of the proposed scheme lie in the fact that the explicit finite difference approximate inverse matrix used as smoother in the multigrid method is a preconditioner for specific sparsity pattern. Further research is carried out in order to derive a generic explicit approximate inverse for any type of sparsity pattern. Originality/value – A novel smoother for the geometric multigrid method is proposed, based on optimized banded approximate inverse matrix preconditioner, the Richardson method in conjunction with the DOUR scheme, for solving large sparse linear systems derived from finite difference discretization of PDEs. Moreover, the applicability and convergence behavior of the proposed scheme is examined based on various cycles and comparative results are given against the damped Jacobi smoother.

Reduction of Linear Systems of ODEs with Optimal Replacement Variables

2011 ◽  
Vol 9 (3) ◽  
pp. 756-779
Author(s):  
Alex Solomonoff ◽  
Wai Sun Don
Keyword(s):  
Linear Systems ◽  
Ad Hoc ◽  
Original System ◽  
Linear Combinations ◽  
Optimal Replacement ◽  
Small Set ◽  
Systems Of Odes ◽  
The Difference ◽  
Linear Odes ◽  
Linear Pdes

AbstractIn this exploratory study, we present a new method of approximating a large system of ODEs by one with fewer equations, while attempting to preserve the essential dynamics of a reduced set of variables of interest. The method has the following key elements: (i) put a (simple, ad-hoc) probability distribution on the phase space of the ODE; (ii) assert that a small set of replacement variables are to be unknown linear combinations of the not-of-interest variables, and let the variables of the reduced system consist of the variables-of-interest together with the replacement variables; (iii) find the linear combinations that minimize the difference between the dynamics of the original system and the reduced system. We describe this approach in detail for linear systems of ODEs. Numerical techniques and issues for carrying out the required minimization are presented. Examples of systems of linear ODEs and variable-coefficient linear PDEs are used to demonstrate the method. We show that the resulting approximate reduced system of ODEs gives good approximations to the original system. Finally, some directions for further work are outlined.

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