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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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