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Volume 2, Issue 4
Taylor Expansion Algorithm for the Branching Solution of the Navier-Stokes Equations

K. Li & Y. He

Int. J. Numer. Anal. Mod., 2 (2005), pp. 459-478.

Published online: 2005-02

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  • Abstract

The aim of this paper is to present a general algorithm for the branching solution of nonlinear operator equations in a Hilbert space, namely the $k$-order Taylor expansion algorithm, $k \geq 1$. The standard Galerkin method can be viewed as the 1-order Taylor expansion algorithm; while the optimum nonlinear Galerkin method can be viewed as the 2-order Taylor expansion algorithm. The general algorithm is then applied to the study of the numerical approximations for the steady Navier-Stokes equations. Finally, the theoretical analysis and numerical experiments show that, in some situations, the optimum nonlinear Galerkin method provides higher convergence rate than the standard Galerkin method and the nonlinear Galerkin method.

  • Keywords

nonlinear operator equation, the Navier-Stokes equations, Taylor expansion algorithm, optimum nonlinear Galerkin method.

  • AMS Subject Headings

35Q10, 47H, 65N30, 76D05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-2-459, author = {K. Li and Y. He}, title = {Taylor Expansion Algorithm for the Branching Solution of the Navier-Stokes Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2005}, volume = {2}, number = {4}, pages = {459--478}, abstract = {

The aim of this paper is to present a general algorithm for the branching solution of nonlinear operator equations in a Hilbert space, namely the $k$-order Taylor expansion algorithm, $k \geq 1$. The standard Galerkin method can be viewed as the 1-order Taylor expansion algorithm; while the optimum nonlinear Galerkin method can be viewed as the 2-order Taylor expansion algorithm. The general algorithm is then applied to the study of the numerical approximations for the steady Navier-Stokes equations. Finally, the theoretical analysis and numerical experiments show that, in some situations, the optimum nonlinear Galerkin method provides higher convergence rate than the standard Galerkin method and the nonlinear Galerkin method.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/941.html} }
TY - JOUR T1 - Taylor Expansion Algorithm for the Branching Solution of the Navier-Stokes Equations AU - K. Li & Y. He JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 459 EP - 478 PY - 2005 DA - 2005/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/941.html KW - nonlinear operator equation, the Navier-Stokes equations, Taylor expansion algorithm, optimum nonlinear Galerkin method. AB -

The aim of this paper is to present a general algorithm for the branching solution of nonlinear operator equations in a Hilbert space, namely the $k$-order Taylor expansion algorithm, $k \geq 1$. The standard Galerkin method can be viewed as the 1-order Taylor expansion algorithm; while the optimum nonlinear Galerkin method can be viewed as the 2-order Taylor expansion algorithm. The general algorithm is then applied to the study of the numerical approximations for the steady Navier-Stokes equations. Finally, the theoretical analysis and numerical experiments show that, in some situations, the optimum nonlinear Galerkin method provides higher convergence rate than the standard Galerkin method and the nonlinear Galerkin method.

K. Li and Y. He. (2005). Taylor Expansion Algorithm for the Branching Solution of the Navier-Stokes Equations. International Journal of Numerical Analysis and Modeling. 2 (4). 459-478. doi:
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