Convergency of iterative methods for implicit solution schemes of fluid flows.

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  • 陰解法に基づく流れの数値解法における反復解法の収束性
  • インカイホウ ニ モトズク ナガレ ノ スウチ カイホウ ニ オケル ハンプク

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Abstract

Implicit difference schemes have been regarded as unconditionally stable methods. Iterative methods are however usually adopted to solve simultaneous equations resulting from the implicit schemes. If the iteration fails to converge, the unconditional stability cannot be obtained in practice. In the present study, convergency of the iterative methods for the implicit schemes with space-centered differencing was examined.<BR>First, the convergency of linear iterative methods for the linear transport equation was analyzed with the spectral radius of the iteration matrix. This showed that the iteration diverges unless the Courant number less than about unity.<BR>Next, numerical experiments were conducted to check the convergency of four iterative methods for the solution of the nonlinear simultaneous equation derived from Burgers' equation. It was confirmed that the iterative methods break the unconditional stability of the implicit scheme, except for the case that the direct methods are applied to solve the linearized equation obtained by the Newton method.<BR>Lastly, the shock tube problem was solved using the four iterative methods. As a result, the unconditional stabitity could not be obtained. This experiment confirmed that the convergency is subject to the Courant condition, the Courant-Friedricks-Levy condition or the degree of discontinuity in state variables.

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