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Algorithms for the exact solution of two matrix equations occurring in the driven cavity problem

Klonowska M. E., Prosnak W. J.

Oceanologia

1999

No. 41 (1)

3-23

It is shown that two particular systems of linear equations, derived in an earlier paper by Prosnak and Kosma (1991), can be solved in an exact time- and storage-saving manner. First of all, by the proper elimination of unknowns, each system can be reduced to a smaller one containing only half of the unknowns. In the first case, the matrix of coefficients of the so reduced system turns out to be tridiagonal, its elements consisting of square submatrices. Moreover, the reduced system can be split into two independent ones. In the second case, the matrix of the reduced system can be presented as the product of two triangular ones, each one being partitioned in square submatrices. Corresponding algorithms and computer programs have been developed in order to investigate whether some economy in storage and computing time is really attainable. Affirmative conclusions are drawn from the results of computations. This means that the new method of solving problems governed by the Navier-Stokes equations, presented in the cited paper, can be applied in a more effective manner.

Tests on a new method for numerical solution to the Navier-Stokes equations*. Part I: plane unsteady flow in finite, rectangular domains

Klonowska M. E., Kołodziejczyk M.

Applied Mechanics and Engineering

1999

Vol. 4, no 2

311-339

The paper deals with a new method (Prosnak and Kosma, 1991) for the determination of unsteady, plane flows of viscous incompressible fluids. The characteristic feature of the method consists in elimination of pressure from the system of the Navier-Stokes equations governing the flow - in such a manner that the order of the resulting system is not increased in comparison with the original one. Furthermore, the mathematical problem posed for the resulting system is reduced in the frame of the method to an initial problem for a system of first order ordinary differential equations, wherein time represents the only independent variable. The nonlinearities of the Navier-Stokes equations do not cause any difficulties by virtue of such an approach. In this paper, the method has been applied to flows in plane, finite rectangular domains, and domains composed of rectangles. Numerical solutions to such problems are presented in the paper in graphical form, and some conclusions are drawn concerning the results as well as the method.