Volume 11, Issue 4
Fixing the Residual Flattening of an Upwind Compact Scheme for Steady Incompressible Flows in Enclosed Domains

Yunchu Wang & Li Yuan

East Asian J. Appl. Math., 11 (2021), pp. 708-731.

Published online: 2021-08

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  • Abstract

The iterative convergence of the upwind compact finite difference scheme for the artificial compressibility method [A. Shah et al, A third-order upwind compact scheme on curvilinear meshes for the incompressible Navier-Stokes equations, Commun. Comput. Phys. 5 (2009)] is studied. It turns out that for steady flows in enclosed domains the residuals do not converge to machine zero. The reason is a non-uniqueness of the calculated pressure in the case where Neumann boundary conditions for the pressure are imposed on all boundaries. The problem can be fixed by modifying the derivatives of mass flux obtained from the upwind compact scheme to satisfy the global mass conservation constraint. Numerical tests show that with this modification the scheme converges to machine zero with the original third-order accuracy.

  • Keywords

Incompressible Navier-Stokes equations, artificial compressibility, upwind compact difference, convergence, enclosed domain.

  • AMS Subject Headings

76D05, 65M06

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

lyuan@lsec.cc.ac.cn (Li Yuan)

  • BibTex
  • RIS
  • TXT
@Article{EAJAM-11-708, author = {Yunchu and Wang and and 17757 and and Yunchu Wang and Li and Yuan and lyuan@lsec.cc.ac.cn and 17758 and ICMSEC and LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China. and Li Yuan}, title = {Fixing the Residual Flattening of an Upwind Compact Scheme for Steady Incompressible Flows in Enclosed Domains}, journal = {East Asian Journal on Applied Mathematics}, year = {2021}, volume = {11}, number = {4}, pages = {708--731}, abstract = {

The iterative convergence of the upwind compact finite difference scheme for the artificial compressibility method [A. Shah et al, A third-order upwind compact scheme on curvilinear meshes for the incompressible Navier-Stokes equations, Commun. Comput. Phys. 5 (2009)] is studied. It turns out that for steady flows in enclosed domains the residuals do not converge to machine zero. The reason is a non-uniqueness of the calculated pressure in the case where Neumann boundary conditions for the pressure are imposed on all boundaries. The problem can be fixed by modifying the derivatives of mass flux obtained from the upwind compact scheme to satisfy the global mass conservation constraint. Numerical tests show that with this modification the scheme converges to machine zero with the original third-order accuracy.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.281120.060421}, url = {http://global-sci.org/intro/article_detail/eajam/19369.html} }
TY - JOUR T1 - Fixing the Residual Flattening of an Upwind Compact Scheme for Steady Incompressible Flows in Enclosed Domains AU - Wang , Yunchu AU - Yuan , Li JO - East Asian Journal on Applied Mathematics VL - 4 SP - 708 EP - 731 PY - 2021 DA - 2021/08 SN - 11 DO - http://doi.org/10.4208/eajam.281120.060421 UR - https://global-sci.org/intro/article_detail/eajam/19369.html KW - Incompressible Navier-Stokes equations, artificial compressibility, upwind compact difference, convergence, enclosed domain. AB -

The iterative convergence of the upwind compact finite difference scheme for the artificial compressibility method [A. Shah et al, A third-order upwind compact scheme on curvilinear meshes for the incompressible Navier-Stokes equations, Commun. Comput. Phys. 5 (2009)] is studied. It turns out that for steady flows in enclosed domains the residuals do not converge to machine zero. The reason is a non-uniqueness of the calculated pressure in the case where Neumann boundary conditions for the pressure are imposed on all boundaries. The problem can be fixed by modifying the derivatives of mass flux obtained from the upwind compact scheme to satisfy the global mass conservation constraint. Numerical tests show that with this modification the scheme converges to machine zero with the original third-order accuracy.

Yunchu Wang & Li Yuan. (2021). Fixing the Residual Flattening of an Upwind Compact Scheme for Steady Incompressible Flows in Enclosed Domains. East Asian Journal on Applied Mathematics. 11 (4). 708-731. doi:10.4208/eajam.281120.060421
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