A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem

Yanhua Cao; Zhendong Luo

doi:doi:10.11648/j.acm.20180701.12

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A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem

Yanhua Cao, Zhendong Luo

Published in Applied and Computational Mathematics (Volume 7, Issue 1)

Received: 4 December 2017 Accepted: 15 December 2017 Published: 12 January 2018

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Abstract

In this study, a stable and convergent finite difference (FD) scheme based on staggered meshes for two-dimensional (2D) incompressible nonlinear viscoelastic fluid dynamics problem including the velocity vector field and the pressure field as well as the deformation tensor matrix is established in order to find numerical solutions for the problem. The stability, convergence, and errors of the FD solutions are analyzed. Some numerical experiments are presented to show that the FD scheme is feasible and efficient for simulating the phenomena of the velocity and the pressure as well as the deformation tensor in an estuary.

Published in	Applied and Computational Mathematics (Volume 7, Issue 1)
DOI	10.11648/j.acm.20180701.12
Page(s)	11-18
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Finite Difference Scheme, Incompressible Nonlinear Viscoelastic Fluid Problem, Stability and Convergence, Numerical Simulations

References

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[7]	I. E. Barton and R. Kirby, “Finite difference scheme for the solution of fluid flow problems on non-staggered grids,” International Journal for Numerical Methods in Fluids, 2000, 33(7), 939–959.
[8]	M. M. Khader and A. M. Megahed, “Numerical simulation using the finite difference method for the flow and heat transfer in a thin liquid film over an unsteady stretching sheet in a saturated porous medium in the presence of thermal radiation,” Journal of King Saud University-Engineering Sciences, 2013, 25, 29–34.
[9]	S. K. Pandit, J. C. Kalita and D. C. Dalal, “A transient higher order compact scheme for incompressible viscous flows on geometries beyond rectangular,” Journal of Computational Physics, 2007, 225(1), 1100–1124.
[10]	I. Soukup, “Numerical method based on DGM for solving the system of equations describing motion of viscoelastic fluid with memory,” In: Numerical Mathematics and Advanced Applications ENUMATH 2015, B. Karasözen et al. (eds.). Lecture Notes in Computational Science and Engineering, Springer International Publishing Switzerland, 2016, 112, 205–213.
[11]	P. A. B. de Sampaio, “A stabilized finite element method for incompressible flow and heat transfer: A natural derivation based on the use of local time-steps,” Computer Methods in Applied Mechanics and Engineering, 2006, 195(44-47), 6177–6190.
[12]	P. A. B. de Sampaio, “A finite element formulation for transient incompressible viscous flows stabilized by local time-steps,” Computer Methods in Applied Mechanics and Engineering, 2005, 194(18-20), 2095–2108.
[13]	C. O. Faria, and J. Karam-Filho, “A regularized-stabilized mixed finite element formulation for viscoplasticity of Bingham type,” Computers & Mathematics with Applications, 2013, 66(6), 975-995.
[14]	H. Xia and Z. D. Luo, “Stabilized finite volume element method for the 2D nonlinear incompressible viscoelastic flow equation,” Boundary Value Problems, 2017, 2017(130), 1–17.
[15]	T. Chung, “Computational Fluid Dynamics”, Cambridge University Press, Cambridge, 2002.
[16]	R. X. Liu, and C. W. Shu, “Several New Methods for Computational Fluid Mechanics (in Chinese),” Science Press, Beijing, 2003.

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APA Style

Yanhua Cao, Zhendong Luo. (2018). A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem. Applied and Computational Mathematics, 7(1), 11-18. https://doi.org/10.11648/j.acm.20180701.12

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ACS Style

Yanhua Cao; Zhendong Luo. A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem. Appl. Comput. Math. 2018, 7(1), 11-18. doi: 10.11648/j.acm.20180701.12

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AMA Style

Yanhua Cao, Zhendong Luo. A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem. Appl Comput Math. 2018;7(1):11-18. doi: 10.11648/j.acm.20180701.12

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@article{10.11648/j.acm.20180701.12,
  author = {Yanhua Cao and Zhendong Luo},
  title = {A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem},
  journal = {Applied and Computational Mathematics},
  volume = {7},
  number = {1},
  pages = {11-18},
  doi = {10.11648/j.acm.20180701.12},
  url = {https://doi.org/10.11648/j.acm.20180701.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180701.12},
  abstract = {In this study, a stable and convergent finite difference (FD) scheme based on staggered meshes for two-dimensional (2D) incompressible nonlinear viscoelastic fluid dynamics problem including the velocity vector field and the pressure field as well as the deformation tensor matrix is established in order to find numerical solutions for the problem. The stability, convergence, and errors of the FD solutions are analyzed. Some numerical experiments are presented to show that the FD scheme is feasible and efficient for simulating the phenomena of the velocity and the pressure as well as the deformation tensor in an estuary.},
 year = {2018}
}

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TY  - JOUR
T1  - A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem
AU  - Yanhua Cao
AU  - Zhendong Luo
Y1  - 2018/01/12
PY  - 2018
N1  - https://doi.org/10.11648/j.acm.20180701.12
DO  - 10.11648/j.acm.20180701.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 11
EP  - 18
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20180701.12
AB  - In this study, a stable and convergent finite difference (FD) scheme based on staggered meshes for two-dimensional (2D) incompressible nonlinear viscoelastic fluid dynamics problem including the velocity vector field and the pressure field as well as the deformation tensor matrix is established in order to find numerical solutions for the problem. The stability, convergence, and errors of the FD solutions are analyzed. Some numerical experiments are presented to show that the FD scheme is feasible and efficient for simulating the phenomena of the velocity and the pressure as well as the deformation tensor in an estuary.
VL  - 7
IS  - 1
ER  -

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Author Information

Yanhua Cao

School of Sciences, East China Jiaotong University, Nanchang, China
Zhendong Luo

School of Mathematics and Physics, North China Electric Power University, Beijing, China

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APA Style

Yanhua Cao, Zhendong Luo. (2018). A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem. Applied and Computational Mathematics, 7(1), 11-18. https://doi.org/10.11648/j.acm.20180701.12

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ACS Style

Yanhua Cao; Zhendong Luo. A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem. Appl. Comput. Math. 2018, 7(1), 11-18. doi: 10.11648/j.acm.20180701.12

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AMA Style

Yanhua Cao, Zhendong Luo. A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem. Appl Comput Math. 2018;7(1):11-18. doi: 10.11648/j.acm.20180701.12

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@article{10.11648/j.acm.20180701.12,
  author = {Yanhua Cao and Zhendong Luo},
  title = {A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem},
  journal = {Applied and Computational Mathematics},
  volume = {7},
  number = {1},
  pages = {11-18},
  doi = {10.11648/j.acm.20180701.12},
  url = {https://doi.org/10.11648/j.acm.20180701.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180701.12},
  abstract = {In this study, a stable and convergent finite difference (FD) scheme based on staggered meshes for two-dimensional (2D) incompressible nonlinear viscoelastic fluid dynamics problem including the velocity vector field and the pressure field as well as the deformation tensor matrix is established in order to find numerical solutions for the problem. The stability, convergence, and errors of the FD solutions are analyzed. Some numerical experiments are presented to show that the FD scheme is feasible and efficient for simulating the phenomena of the velocity and the pressure as well as the deformation tensor in an estuary.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem
AU  - Yanhua Cao
AU  - Zhendong Luo
Y1  - 2018/01/12
PY  - 2018
N1  - https://doi.org/10.11648/j.acm.20180701.12
DO  - 10.11648/j.acm.20180701.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 11
EP  - 18
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20180701.12
AB  - In this study, a stable and convergent finite difference (FD) scheme based on staggered meshes for two-dimensional (2D) incompressible nonlinear viscoelastic fluid dynamics problem including the velocity vector field and the pressure field as well as the deformation tensor matrix is established in order to find numerical solutions for the problem. The stability, convergence, and errors of the FD solutions are analyzed. Some numerical experiments are presented to show that the FD scheme is feasible and efficient for simulating the phenomena of the velocity and the pressure as well as the deformation tensor in an estuary.
VL  - 7
IS  - 1
ER  -

Copy | Download