Efficient and Unconditional Energy Stable Schemes for the Micropolar Navier-Stokes Equations

Abstract

We develop in this paper efficient numerical schemes for solving the micropolar Navier-Stokes equations by combining the SAV approach and pressure-correction method. Our first- and second-order semi-discrete schemes enjoy remarkable properties such as (i) unconditional energy stability with a modified energy, and (ii) only a sequence of decoupled linear equations with constant coefficients need to be solved at each time step. We also construct fully discrete versions of these schemes with a special spectral discretization which preserve the essential properties of the semi-discrete schemes. Numerical experiments are presented to validate the proposed schemes.

Keywords

Micropolar Navier-Stokes / pressure-correction / scalar auxiliary variable / energy stability

References

[1] E. Ahusborde, M. Azaiez, and J.-P. Caltagirone. A primal formulation for the Helmholtz decomposition. J. Comput. Phys., 225(1):13-19, 2007.

[2] C. Bernardi and Y. Maday. Spectral methods. In Handbook of Numerical Analysis, Vol. V, Handb. Numer. Anal., V, pages 209-485. North-Holland, Amsterdam, 1997.

[3] C. Bernardi, Y. Maday, and F. Rapetti. Discrétisations variationnelles de problèmes aux limites elliptiques, volume 45 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin, 2004.

[4] F. Chen and J. Shen. Efficient spectral-Galerkin methods for systems of coupled secondorder equations and their applications. J. Comput. Phys., 231(15):5016-5028, 2012.

[5] B.-Q. Dong and Z. Zhang. Global regularity of the 2D micropolar fluid flows with zero angular viscosity. Journal of Differential Equations, 249(1):200-213, 2010.

[6] A. C. Eringen. Theory of micropolar fluids. Journal ofMathematics andMechanics, pages 1-18, 1966.

[7] C. Ferrari and R. Gilbert. On lubrication with structured fluids. Applicable Analysis, 15(1-4):127-146, 1983.

[8] G. P. Galdi and S. Rionero. A note on the existence and uniqueness of solutions of the micropolar fluid equations. International Journal of Engineering Science, 15(2):105-108, 1977.

[9] J. Guermond and J. Shen. On the error estimates for the rotational pressure-correction projection methods. Mathematics of Computation, 73(248):1719-1737.

[10] J.-L. Guermond, P. Minev, and J. Shen. An overview of projection methods for incompressible flows. Computer Methods in Applied Mechanics and Engineering, 195(44-47):6011-6045, 2006.

[11] X. Li, J. Shen, and Z. Liu. New SAV-pressure correction methods for the Navier-Stokes equations: Stability and error analysis. arXiv preprint arXiv:2002. 09090, 2020.

[12] L. Lin, Z. Yang, and S. Dong. Numerical approximation of incompressible Navier-Stokes equations based on an auxiliary energy variable. J. Comput. Phys., 388:1-22, 2019.

[13] G. Lukaszewicz. Micropolar Fluids: Theory and Applications. Springer Science & Business Media, 1999.

[14] G. Łukaszewicz. Long time behavior of 2D micropolar fluid flows. Mathematical and Computer Modelling, 34(5-6):487-509, 2001.

[15] R. H. Nochetto, A. J. Salgado, and I. Tomas. The micropolar Navier-Stokes equations: a priori error analysis. Mathematical Models and Methods in Applied Sciences, 24(07):1237-1264, 2014.

[16] R.H.Nochetto, A. J. Salgado, and I. Tomas. The equations of ferrohydrodynamics: Modeling and numerical methods. Mathematical Models and Methods in Applied Sciences, 26(13):2393-2449, 2016.

[17] E. Ortega-Torres and M. Rojas-Medar. Optimal error estimate of the penalty finite element method for the micropolar fluid equations. Numerical Functional Analysis and Optimization, 29(5-6):612-637, 2008.

[18] E. E. Ortega-Torres and M. A. Rojas-Medar. Magneto-micropolar fluid motion: Global existence of strong solutions. In Abstract and Applied Analysis, volume 4. Hindawi, 1999.

[19] J.-H. Pyo. Error estimates for the second order semi-discrete stabilized Gauge-Uzawa method for the Navier-Stokes equations. International Journal of Numerical Analysis & Modeling, 10(1), 2013.

[20] M. A. Rojas-Medar. Magneto-micropolar fluid motion: Existence and uniqueness of strong solution. Mathematische Nachrichten, 188(1):301-319, 1997.

[21] A. J. Salgado. Convergence analysis of fractional time-stepping techniques for incompressible fluids with microstructure. Journal of Scientific Computing, 64(1):216-233, 2015.

[22] V. U. K. Sastry and T. Das. Stability of Couette flow and Dean flow in micropolar fluids. International Journal of Engineering Science, 23(11):1163-1177, 1985.

[23] J. Shen, T. Tang, and L.-L. Wang. Spectral Methods, volume 41 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2011. Algorithms, Analysis and Applications.

[24] J. Shen, J. Xu, and J. Yang. A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev., 61(3):474-506, 2019.