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Volume 15, Issue 3
An HOC Finite Difference Scheme for the Steady Natural Convection Problem Based on the Velocity-Vorticity Method

Tao Wang, Tiegang Liu & Kun Wang

East Asian J. Appl. Math., 15 (2025), pp. 591-614.

Published online: 2025-06

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  • Abstract

A high-order compact finite difference scheme for solving natural convection problems using velocity-vorticity formulation of the incompressible Navier-Stokes equations is presented. The basic idea of the method is to regard all controlling equations as the Poisson-type. We construct a fourth-order finite difference scheme for the velocity-vorticity equation based on the nine-point stencils for each Poisson-type equation. Next we give an example with an exact solution to verify that the scheme has the fourth-order accuracy. Finally, numerical solutions for the model problem of natural convection in a square heating cavity are presented to show the reliability and effectiveness of this method.

  • AMS Subject Headings

65Y04, 65Z05, 65N06, 65D25

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-15-591, author = {Wang , TaoLiu , Tiegang and Wang , Kun}, title = {An HOC Finite Difference Scheme for the Steady Natural Convection Problem Based on the Velocity-Vorticity Method}, journal = {East Asian Journal on Applied Mathematics}, year = {2025}, volume = {15}, number = {3}, pages = {591--614}, abstract = {

A high-order compact finite difference scheme for solving natural convection problems using velocity-vorticity formulation of the incompressible Navier-Stokes equations is presented. The basic idea of the method is to regard all controlling equations as the Poisson-type. We construct a fourth-order finite difference scheme for the velocity-vorticity equation based on the nine-point stencils for each Poisson-type equation. Next we give an example with an exact solution to verify that the scheme has the fourth-order accuracy. Finally, numerical solutions for the model problem of natural convection in a square heating cavity are presented to show the reliability and effectiveness of this method.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2024-056.270624}, url = {http://global-sci.org/intro/article_detail/eajam/24157.html} }
TY - JOUR T1 - An HOC Finite Difference Scheme for the Steady Natural Convection Problem Based on the Velocity-Vorticity Method AU - Wang , Tao AU - Liu , Tiegang AU - Wang , Kun JO - East Asian Journal on Applied Mathematics VL - 3 SP - 591 EP - 614 PY - 2025 DA - 2025/06 SN - 15 DO - http://doi.org/10.4208/eajam.2024-056.270624 UR - https://global-sci.org/intro/article_detail/eajam/24157.html KW - Navier-Stokes equation, Boussinesq hypothesis, velocity-vorticity method, fourth-order compact scheme, natural convection problem. AB -

A high-order compact finite difference scheme for solving natural convection problems using velocity-vorticity formulation of the incompressible Navier-Stokes equations is presented. The basic idea of the method is to regard all controlling equations as the Poisson-type. We construct a fourth-order finite difference scheme for the velocity-vorticity equation based on the nine-point stencils for each Poisson-type equation. Next we give an example with an exact solution to verify that the scheme has the fourth-order accuracy. Finally, numerical solutions for the model problem of natural convection in a square heating cavity are presented to show the reliability and effectiveness of this method.

Wang , TaoLiu , Tiegang and Wang , Kun. (2025). An HOC Finite Difference Scheme for the Steady Natural Convection Problem Based on the Velocity-Vorticity Method. East Asian Journal on Applied Mathematics. 15 (3). 591-614. doi:10.4208/eajam.2024-056.270624
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