On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals
In the work, we consider the problem of accelerating the iteration process of the numerical solution of boundary-value problems for partial differential equations (PDE) by the method of collocations and least residuals (CLR). To solve this problem, it is proposed to combine simultaneously three tech...
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Yaroslavl State University
2017-02-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/425 |
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| author | Vasily P. Shapeev Evgenii V. Vorozhtsov |
| author_facet | Vasily P. Shapeev Evgenii V. Vorozhtsov |
| author_sort | Vasily P. Shapeev |
| collection | DOAJ |
| description | In the work, we consider the problem of accelerating the iteration process of the numerical solution of boundary-value problems for partial differential equations (PDE) by the method of collocations and least residuals (CLR). To solve this problem, it is proposed to combine simultaneously three techniques of the iteration process acceleration: the preconditioner, the multigrid algorithm, and the correction of the PDE solution at the intermediate iterations in the Krylov subspace. The influence of all three techniques of the iteration acceleration was investigated both individually for each technique and at their combination. Each of the above techniques is shown to make its contribution to the quantitative figure of iteration process speed-up. The algorithm which employs the Krylov subspaces makes the most significant contribution. The joint simultaneous application of all three techniques for accelerating the iterative solution of specific boundary-value problems enabled a reduction of the CPU time of their solution on computer by a factor of up to 230 in comparison with the case when no acceleration techniques were applied. A two-parameter preconditioner was investigated. It is proposed to find the optimal values of its parameters by the numerical solution of a computationally inexpensive problem of minimizing the condition number of the system of linear algebraic equations. The problem is solved by the CLR method and it is modified by the preconditioner. It is shown that it is sufficient to restrict oneself in the multigrid version of the CLR method only to a simple solution prolongation operation on the multigrid complex to reduce substantially the CPU time of the boundary-value problem solution.Numerous computational examples are presented, which demonstrate the efficiency of the approaches proposed for accelerating the iterative processes of the numerical solution of the boundary-value problems for the two-dimensional Navier–Stokes equations. It is pointed out that the proposed combination of the techniques for accelerating the iteration processes may be also implemented within the framework of other numerical techniques for the solution of PDEs. |
| format | Article |
| id | doaj-art-758e1c60f3284f06b24594e7b033fa82 |
| institution | Kabale University |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2017-02-01 |
| publisher | Yaroslavl State University |
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| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-758e1c60f3284f06b24594e7b033fa822025-08-20T03:37:44ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172017-02-01241396310.18255/1818-1015-2017-1-39-63352On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least ResidualsVasily P. Shapeev0Evgenii V. Vorozhtsov1Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of Sciences; Novosibirsk National Research UniversityKhristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of the Russian Academy of SciencesIn the work, we consider the problem of accelerating the iteration process of the numerical solution of boundary-value problems for partial differential equations (PDE) by the method of collocations and least residuals (CLR). To solve this problem, it is proposed to combine simultaneously three techniques of the iteration process acceleration: the preconditioner, the multigrid algorithm, and the correction of the PDE solution at the intermediate iterations in the Krylov subspace. The influence of all three techniques of the iteration acceleration was investigated both individually for each technique and at their combination. Each of the above techniques is shown to make its contribution to the quantitative figure of iteration process speed-up. The algorithm which employs the Krylov subspaces makes the most significant contribution. The joint simultaneous application of all three techniques for accelerating the iterative solution of specific boundary-value problems enabled a reduction of the CPU time of their solution on computer by a factor of up to 230 in comparison with the case when no acceleration techniques were applied. A two-parameter preconditioner was investigated. It is proposed to find the optimal values of its parameters by the numerical solution of a computationally inexpensive problem of minimizing the condition number of the system of linear algebraic equations. The problem is solved by the CLR method and it is modified by the preconditioner. It is shown that it is sufficient to restrict oneself in the multigrid version of the CLR method only to a simple solution prolongation operation on the multigrid complex to reduce substantially the CPU time of the boundary-value problem solution.Numerous computational examples are presented, which demonstrate the efficiency of the approaches proposed for accelerating the iterative processes of the numerical solution of the boundary-value problems for the two-dimensional Navier–Stokes equations. It is pointed out that the proposed combination of the techniques for accelerating the iteration processes may be also implemented within the framework of other numerical techniques for the solution of PDEs.https://www.mais-journal.ru/jour/article/view/425preconditioningkrylov subspacesmultigridgauss–seidel iterationsnavier–stokes equationsthe method of collocations and least residuals |
| spellingShingle | Vasily P. Shapeev Evgenii V. Vorozhtsov On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals Моделирование и анализ информационных систем preconditioning krylov subspaces multigrid gauss–seidel iterations navier–stokes equations the method of collocations and least residuals |
| title | On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals |
| title_full | On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals |
| title_fullStr | On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals |
| title_full_unstemmed | On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals |
| title_short | On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals |
| title_sort | on combining different acceleration techniques at the iterative solution of pdes by the method of collocations and least residuals |
| topic | preconditioning krylov subspaces multigrid gauss–seidel iterations navier–stokes equations the method of collocations and least residuals |
| url | https://www.mais-journal.ru/jour/article/view/425 |
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