Volume 12, Issue 1
A Consistent Fourth-Order Compact Finite Difference Scheme for Solving Vorticity-Stream Function Form of Incompressible Navier-Stokes Equations

Tao Wang & Tiegang Liu

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 312-330.

Published online: 2018-09

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  • Abstract

The inconsistent accuracy and truncation error in the treatment of boundary usually leads to performance defects, such as decreased accuracy and even numerical instability, of the entire computational method, especially for higher order methods. In this work, we construct a consistent fourth-order compact finite difference scheme for solving two-dimensional incompressible Navier-Stokes (N-S) equations. In the proposed method, the main truncation error term of the boundary scheme is kept the same as that of the interior compact finite difference scheme. With such a feature, the numerical stability and accuracy of the entire computation can be maintained the same as the interior compact finite difference scheme. Numerical examples show the effectiveness and accuracy of the present consistent compact high order scheme in $L$. Its application to two dimensional lid-driven cavity flow problem further exhibits that under the same condition, the computed solution with the present scheme is much close to the benchmark in comparison to those from the 4$th$ order explicit scheme. The compact finite difference method equipped with the present consistent boundary technique improves much the stability of the whole computation and shows its potential application to incompressible flow of high Reynolds number.

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@Article{NMTMA-12-312, author = {}, title = {A Consistent Fourth-Order Compact Finite Difference Scheme for Solving Vorticity-Stream Function Form of Incompressible Navier-Stokes Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {12}, number = {1}, pages = {312--330}, abstract = {

The inconsistent accuracy and truncation error in the treatment of boundary usually leads to performance defects, such as decreased accuracy and even numerical instability, of the entire computational method, especially for higher order methods. In this work, we construct a consistent fourth-order compact finite difference scheme for solving two-dimensional incompressible Navier-Stokes (N-S) equations. In the proposed method, the main truncation error term of the boundary scheme is kept the same as that of the interior compact finite difference scheme. With such a feature, the numerical stability and accuracy of the entire computation can be maintained the same as the interior compact finite difference scheme. Numerical examples show the effectiveness and accuracy of the present consistent compact high order scheme in $L$. Its application to two dimensional lid-driven cavity flow problem further exhibits that under the same condition, the computed solution with the present scheme is much close to the benchmark in comparison to those from the 4$th$ order explicit scheme. The compact finite difference method equipped with the present consistent boundary technique improves much the stability of the whole computation and shows its potential application to incompressible flow of high Reynolds number.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0043}, url = {http://global-sci.org/intro/article_detail/nmtma/12702.html} }
TY - JOUR T1 - A Consistent Fourth-Order Compact Finite Difference Scheme for Solving Vorticity-Stream Function Form of Incompressible Navier-Stokes Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 312 EP - 330 PY - 2018 DA - 2018/09 SN - 12 DO - http://doi.org/10.4208/nmtma.OA-2018-0043 UR - https://global-sci.org/intro/article_detail/nmtma/12702.html KW - AB -

The inconsistent accuracy and truncation error in the treatment of boundary usually leads to performance defects, such as decreased accuracy and even numerical instability, of the entire computational method, especially for higher order methods. In this work, we construct a consistent fourth-order compact finite difference scheme for solving two-dimensional incompressible Navier-Stokes (N-S) equations. In the proposed method, the main truncation error term of the boundary scheme is kept the same as that of the interior compact finite difference scheme. With such a feature, the numerical stability and accuracy of the entire computation can be maintained the same as the interior compact finite difference scheme. Numerical examples show the effectiveness and accuracy of the present consistent compact high order scheme in $L$. Its application to two dimensional lid-driven cavity flow problem further exhibits that under the same condition, the computed solution with the present scheme is much close to the benchmark in comparison to those from the 4$th$ order explicit scheme. The compact finite difference method equipped with the present consistent boundary technique improves much the stability of the whole computation and shows its potential application to incompressible flow of high Reynolds number.

Tao Wang & Tiegang Liu. (2020). A Consistent Fourth-Order Compact Finite Difference Scheme for Solving Vorticity-Stream Function Form of Incompressible Navier-Stokes Equations. Numerical Mathematics: Theory, Methods and Applications. 12 (1). 312-330. doi:10.4208/nmtma.OA-2018-0043
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