Volume 12, Issue 5
Unconditional Long Time $\text{H}^1$-Stability of a Velocity-Vorticity-Temperature Scheme for the $2\text{D}$-Boussinesq System

Mine Akbas

Adv. Appl. Math. Mech., 12 (2020), pp. 1166-1195.

Published online: 2020-07

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  • Abstract

This paper proposes, analyzes and tests a velocity-vorticity-temperature (VVT) scheme for incompressible, non-isothermal fluid flow. VVT consists of complementing of the usual velocity-pressure-temperature system with the vorticity equation, coupling the systems through the convective terms. The proposed scheme uses BDF2LE time stepping, and a finite element spatial discretization. At each time step, the velocity-pressure equation, the vorticity equation and the temperature equation are all decoupled. A full analysis of the method is given that proves unconditional long-time $\text{H}^1$-stability, and shows the optimal convergence both in time and space. Theoretical convergence results are confirmed by a numerical test, and the effectiveness of the algorithm is revealed on a benchmark problem for Marsigli flow.

  • Keywords

Long time stability, incompressible flow, vorticity equation, finite element method.

  • AMS Subject Headings

74H40, 76D03, 76D05, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-12-1166, author = {Mine Akbas , }, title = {Unconditional Long Time $\text{H}^1$-Stability of a Velocity-Vorticity-Temperature Scheme for the $2\text{D}$-Boussinesq System}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {5}, pages = {1166--1195}, abstract = {

This paper proposes, analyzes and tests a velocity-vorticity-temperature (VVT) scheme for incompressible, non-isothermal fluid flow. VVT consists of complementing of the usual velocity-pressure-temperature system with the vorticity equation, coupling the systems through the convective terms. The proposed scheme uses BDF2LE time stepping, and a finite element spatial discretization. At each time step, the velocity-pressure equation, the vorticity equation and the temperature equation are all decoupled. A full analysis of the method is given that proves unconditional long-time $\text{H}^1$-stability, and shows the optimal convergence both in time and space. Theoretical convergence results are confirmed by a numerical test, and the effectiveness of the algorithm is revealed on a benchmark problem for Marsigli flow.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0122}, url = {http://global-sci.org/intro/article_detail/aamm/17744.html} }
TY - JOUR T1 - Unconditional Long Time $\text{H}^1$-Stability of a Velocity-Vorticity-Temperature Scheme for the $2\text{D}$-Boussinesq System AU - Mine Akbas , JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1166 EP - 1195 PY - 2020 DA - 2020/07 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0122 UR - https://global-sci.org/intro/article_detail/aamm/17744.html KW - Long time stability, incompressible flow, vorticity equation, finite element method. AB -

This paper proposes, analyzes and tests a velocity-vorticity-temperature (VVT) scheme for incompressible, non-isothermal fluid flow. VVT consists of complementing of the usual velocity-pressure-temperature system with the vorticity equation, coupling the systems through the convective terms. The proposed scheme uses BDF2LE time stepping, and a finite element spatial discretization. At each time step, the velocity-pressure equation, the vorticity equation and the temperature equation are all decoupled. A full analysis of the method is given that proves unconditional long-time $\text{H}^1$-stability, and shows the optimal convergence both in time and space. Theoretical convergence results are confirmed by a numerical test, and the effectiveness of the algorithm is revealed on a benchmark problem for Marsigli flow.

Mine Akbas. (2020). Unconditional Long Time $\text{H}^1$-Stability of a Velocity-Vorticity-Temperature Scheme for the $2\text{D}$-Boussinesq System. Advances in Applied Mathematics and Mechanics. 12 (5). 1166-1195. doi:10.4208/aamm.OA-2019-0122
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