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A realization of the collocation and least squares (CLS) method is reduced to solving of overdetermined system of linear algebraic equations (SLAE). In many papers dedicated to CLS method, for example in [1], SLAE's solution x is found by iterative process x^n, n=0, 1, ... In calculations on fine grids and in other cases a necessity of convergence acceleration appears. There are many acceleration algorithms based on use of Krylov's subspaces [2]. The correction is added to current approach x^n after every k steps in one of them. This correction is a linear combination of equations' residuals r^j for concerned SLAE, where r^j=x^j-x^{j-1}, j=(n-k), ... , (n-1), k < n. An auxiliary overdetermined SLAE L u=f is written for finding this correction in [3], where u is the coefficient vector of sought linear combination. The least squares method (LSM) is used for solving system L u=f in [3]. The realization of LSM is reduced to solving a SLAE with matrix L^T*L. The conditionality of this system is much worse than conditionality of matrix L, which is often ill-conditioned itself. In this paper an orthogonal method for solving system L u=f is proposed and implemented. Solving of the auxiliary system is reduced in the orthodonal method to solving of definite SLAE M u=h, where matrix M is upper-triangular. In the method the exlusion of under-diagonal elements is done by selecting the element with the greatest module. Moreover, the possibility of linearly dependent (or close to linearly dependent) rows presence in matrix L is taken into account in the algorithm. Finally, the maximal non-degenerate submatrix is chosen from matrix L. The solution of SLAE M u=h is close to the solution, which is found by LSM for L u=f. At the same time, the conditionality of matrix M is much better than conditionality of matrix L^T*L.
The convergence acceleration method proposed here was implemented in CLS method for Navier-Stokes equations. The computation time was essentially decreased. In some cases a divergent iterative process becomes convergent after using the acceleration algorithm. These results allow one to use finer grids and to calculate in a wider range of Reynolds numbers Re. As the result, after calculation of the flow in a rectangular lid driven cavity for Re=1000 on the grid 640*640 in this work were found not only known vortexes, but also vortexes of second order of smallness in the cavity's corner. The closest results to the ones obtained here are published in paper [4], where a high-order scheme with a very small artificial viscosity is used. It testifies that the CLS method allows one to obtain a solution with a high accuracy.
The work has been supported by the RFBR, Project No. 06-01-00080-a.
References
[1] Issaev V.I., Shapeev V.P., Eremin S.A. Issledovanije svojstv metoda kollokatsii i naimenshih kvadratov reshenija kraevyh zadach dlya uravnenija Puassona i uravnenij Navje-Stoksa// JCT (in publishing).
[2] Saad Y. Krylov Subspace Methods for Solving Unsymmetric Linear Systems// Mathematics of computation --- 1981. --- Vol. 37, №155 --- P.105--126.
[3] Sleptsov A.G. Ob uskorenii shodimosti linejnyh iteratsij// Modelirovanije v mehanike. --- Novosibirsk, 1989. --- Vol.3(20), №3. --- P.132--147.
[4] Garanzha V.A., Konshin V.N. Chislennyje algoritmy dlya techeniy vyazkoj zhidkosti, osnovannyje na konservativnyh kompaktnyh shemah vysokogo poryadka approksimatsii. JCMMP. --- 1999. --- Vol.39, №8. --- P.1378--1392.
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