Analysis of a fourth order finite difference method for the incompressible Boussinesq equations

Cheng Wang University of Tennessee Jian-Guo Liu University of Maryland Hangs Johnston University of Michigan, Department of Mathematics, USA

Numerical Analysis and Scientific Computing mathscidoc:1702.25045

Numerische Mathematik, 97, (3), 555–594, 2004.5
The convergence of a fourth order finite difference method for the 2-D unsteady, viscous incompressible Boussinesq equations, based on the vorticity-stream function formulation, is established in this article. A compact fourth order scheme is used to discretize the momentum equation, and long-stencil fourth order operators are applied to discretize the temperature transport equation. A local vorticity boundary condition is used to enforce the no-slip boundary condition for the velocity. One-sided extrapolation is used near the boundary, dependent on the type of boundary condition for the temperature, to prescribe the temperature at ‘‘ghost’’ points lying outside of the computational domain. Theoretical results of the stability and accuracy of the method are also provided. In numerical experiments the method has been shown to be capable of producing highly resolved solutions at a reasonable computational cost.
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@inproceedings{cheng2004analysis,
  title={Analysis of a fourth order finite difference method for the incompressible Boussinesq equations},
  author={Cheng Wang, Jian-Guo Liu, and Hangs Johnston},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209101144885407372},
  booktitle={Numerische Mathematik},
  volume={97},
  number={3},
  pages={555–594},
  year={2004},
}
Cheng Wang, Jian-Guo Liu, and Hangs Johnston. Analysis of a fourth order finite difference method for the incompressible Boussinesq equations. 2004. Vol. 97. In Numerische Mathematik. pp.555–594. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209101144885407372.
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