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Efficient Time-Stepping/Spectral Methods for the Navier-Stokes-Nernst-Planck-Poisson Equations

Part of: Basic methods in fluid mechanics Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  27 March 2017

Xiaoling Liu  and
Chuanju Xu
Xiaoling Liu*
Affiliation:
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen University, 361005 Xiamen, China
Chuanju Xu*
Affiliation:
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen University, 361005 Xiamen, China
*
*Corresponding author. Email addresses:liuxiaoling.xmu@163.com (X. Liu), cjxu@xmu.edu.cn (C. Xu)
*Corresponding author. Email addresses:liuxiaoling.xmu@163.com (X. Liu), cjxu@xmu.edu.cn (C. Xu)
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Abstract

This paper is concerned with numerical methods for the Navier-Stokes-Nernst-Planck-Poisson equation system. The main goal is to construct and analyze some stable time stepping schemes for the time discretization and use a spectral method for the spatial discretization. The main contribution of the paper includes: 1) an useful stability inequality for the weak solution is derived; 2) a first order time stepping scheme is constructed, and the non-negativity of the concentration components of the discrete solution is proved. This is an important property since the exact solution shares the same property. Moreover, the stability of the scheme is established, together with a stability condition on the time step size; 3) a modified first order scheme is proposed in order to decouple the calculation of the velocity and pressure in the fluid field. This new scheme equally preserves the non-negativity of the discrete concentration solution, and is stable under a similar stability condition; 4) a stabilization technique is introduced to make the above mentioned schemes stable without restriction condition on the time step size; 5) finally we construct a second order finite difference scheme in time and spectral discretization in space. The numerical tests carried out in the paper show that all the proposed schemes possess some desirable properties, such as conditionally/unconditionally stability, first/second order convergence, non-negativity of the discrete concentrations, and so on.

Keywords

Navier-Stokes-Nernst-Planck-Poisson equationsstabilized finite difference schemesspectral method

MSC classification

Secondary: 76M10: Finite element methods 76M22: Spectral methods 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 5 , May 2017 , pp. 1408 - 1428
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Bazant, Martin Z, Thornton, Katsuyo, and Ajdari, Armand. Diffuse-charge dynamics in electrochemical systems. Physical review E, 70(2):021506, 2004.Google Scholar
[2] Chorin, Alexandre Joel. Numerical solution of the Navier-Stokes equations. Mathematics of computation, 22(104):745–762, 1968.Google Scholar
[3] Deng, Chao, Zhao, Jihong, and Cui, Shangbin. Well-posedness for the Navier–Stokes–Nernst–Planck–Poisson system in Triebel–Lizorkin space and Besov space with negative indices. Journal of Mathematical Analysis and Applications, 377(1):392–405, 2011.Google Scholar
[4] Gilbarg, David and Trudinger, Neil S. Elliptic Partial Differential Equations of Second Order, volume 224. Springer Verlag, 2001.Google Scholar
[5] Girault, Vivette and Raviart, Pierre-Arnaud. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. NASA STI/Recon Technical Report A, 87:52227, 1986.Google Scholar
[6] Grisvard, Pierre. Elliptic Problems in Nonsmooth Domains, volume 69. SIAM, 1985.Google Scholar
[7] Guermond, Jean-Luc, Minev, Peter, and Shen, Jie. An overview of projection methods for incompressible flows. Computer methods in applied mechanics and engineering, 195(44):6011–6045, 2006.Google Scholar
[8] Kirby, Brian. Micro-and nanoscale fluid mechanics: transport in microfluidic devices. Cambridge University Press, 2010.Google Scholar
[9] Probstein, Ronald F. Physicochemical hydrodynamics: an introduction. John Wiley & Sons, 2005.Google Scholar
[10] Prohl, Andreas and Schmuck, Markus. Convergent finite element discretizations of the Navier–Stokes–Nernst–Planck–Poisson system. ESAIM: Mathematical Modelling and Numerical Analysis, 44(3):531–571, 2010.Google Scholar
[11] Quarteroni, Alfio and Valli, Alberto. Numerical Approximation of Partial Differential Equations, 1997.Google Scholar
[12] Reuss, Ferdinand Friedrich. Sur un nouvel effet de l’électricité galvanique. Mem. Soc. Imp. Natur. Moscou, 2:327–337, 1809.Google Scholar
[13] Schmuck, Markus. Analysis of the Navier–Stokes–Nernst–Planck–Poisson system. Mathematical Models and Methods in Applied Sciences, 19(06):993–1014, 2009.Google Scholar
[14] Stryer, Lubert. Biochemistry. W. H. Freeman and Company, New York, 1988.Google Scholar
[15] Temam, Roger. Une méthode d’approximation des solutions des équations de Navier–Stokes. Bull. Soc. Math. France, 98:115–152, 1968.Google Scholar
[16] Tsai, Chien-Hsiung, Yang, Ruey-Jen, Tai, Chang-Hsien, and Fu, Lung-Ming. Numerical simulation of electrokinetic injection techniques in capillary electrophoresis microchips. Electrophoresis, 26(3):674–686, 2005.Google Scholar
[17] Valencia, Lorenzo Héctor Juárez, Flores Rivera, Ciro F, Mora, Eduardo Ramos, and Nunez, José. Fuel Cells: Navier–Stokes and Poisson–Nernst–Planck Equations. I WAPHS, page 137, 2008.Google Scholar
[18] Yang, R-J, Fu, L-M, and Hwang, C-C. Electroosmotic entry flow in a microchannel. Journal of Colloid and Interface Science, 244(1):173–179, 2001.Google Scholar