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Exponential Time Differencing Gauge Method for Incompressible Viscous Flows

Part of: Computer aspects of numerical algorithms Partial differential equations, initial value and time-dependent initial-boundary value problems Incompressible viscous fluids

Published online by Cambridge University Press:  21 June 2017

Lili Ju  and
Zhu Wang
Lili Ju*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
Zhu Wang*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
*
*Corresponding author. Email addresses:ju@math.sc.edu (L. Ju), wangzhu@math.sc.edu (Z.Wang)
*Corresponding author. Email addresses:ju@math.sc.edu (L. Ju), wangzhu@math.sc.edu (Z.Wang)
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Abstract

In this paper, we study an exponential time differencing method for solving the gauge system of incompressible viscous flows governed by Stokes or Navier-Stokes equations. The momentum equation is decoupled from the kinematic equation at a discrete level and is then solved by exponential time stepping multistep schemes in our approach. We analyze the stability of the proposed method and rigorously prove that the first order exponential time differencing scheme is unconditionally stable for the Stokes problem. We also present a compact representation of the algorithm for problems on rectangular domains, which makes FFT-based solvers available for the resulting fully discretized system. Various numerical experiments in two and three dimensional spaces are carried out to demonstrate the accuracy and stability of the proposed method.

Keywords

Incompressible flowsStokes equationsNavier-Stokes equationsgauge methodexponential time differencing

MSC classification

Secondary: 65M06: Finite difference methods 65M22: Solution of discretized equations 65Y20: Complexity and performance of numerical algorithms 76D05: Navier-Stokes equations 76D07: Stokes and related (Oseen, etc.) flows
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 2 , August 2017 , pp. 517 - 541
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Brown, D. L., Cortez, R. and Minion, M. L., Accurate projection methods for the incompressible Navier–Stokes equations, J. Comput. Phys., 168(2), pp. 464–499, 2001.Google Scholar
[2] Calvo, M. P., Palencia, C., A class of explicit multistep exponential integrators for semilinear problems, Numer. Math., 102, pp. 367–381, 2006.Google Scholar
[3] Cox, S. and Matthews, P., Exponential time differencing for stiff systems, J. Comput. Phys., 176, pp. 430–455, 2002.CrossRefGoogle Scholar
[4] Chorin, A. J., Numerical solution of the Navier-Stokes equations, Math. Comp., 22, pp. 745–762, 1968.Google Scholar
[5] Erturk, E., Comparison of wide and compact fourth-order formulations of the Navier-Stokes equations, Inter. J. Numer. Meth. Fluids, 60, pp. 992–1010, 2009.CrossRefGoogle Scholar
[6] W. E, and Liu, J.-G., Finite difference schemes for incompressible flows in the velocity-impulse density formulation, J. Comput. Phys., 130, pp. 67–76, 1997.Google Scholar
[7] W. E, and Liu, J.-G., Gauge finite element method for incompressible flows, Int. J. Num. Meth. Fluids, 34, pp. 701–710, 2000.Google Scholar
[8] W. E, and Liu, J.-G., Gauge method for viscous incompressible flows, Comm. Math. Sci., 1, pp. 317–332, 2003.Google Scholar
[9] Gallopoulos, E. and Saad, Y., Efficient solution of parabolic equations by Krylov approximation methods, SIAM J. Sci. Stat. Comput., 13, pp. 1236–1264, 1992.Google Scholar
[10] Golub, G., Huang, L.-C., Simon, H., and Tang, W.-P., A fast Poisson solver for the finite difference solution of the incompressible Navier-Stokes equations, SIAM J. Sci. Comput., 19, pp. 1606–1624, 1998.Google Scholar
[11] Guermond, J. L., Minev, P. and Shen, J., An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195, pp. 6011–6045, 2006.Google Scholar
[12] Guermond, J. L. and Salgado, A., A splitting method for incompressible flows with variable density based on a pressure Poisson equation, J. Comput. Phys., 228(8), pp. 2834–2846, 2009.Google Scholar
[13] Guermond, J. L. and Salgado, A., Error analysis of a fractional time-stepping technique for incompressible flows with variable density, SIAM J. Numer. Anal., 49(3), pp. 917–944, 2011.Google Scholar
[14] Gunzburger, M. D., Perspectives in Flow Control and Optimization, Society for Industrial and Applied Mathematics, 2003.Google Scholar
[15] Gunzburger, M. D., Finite Element Methods for Viscous Incompressible Flows: a guide to theory, practice, and algorithms, Academic Press, INC., 1989.Google Scholar
[16] Hochbrucky, M. and Lubich, C., On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34, pp. 1911–1925, 1997.Google Scholar
[17] Hochbruck, M., Lubich, C. and Selhofer, H., Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19, pp. 1552–1574, 1998.Google Scholar
[18] Hochbruck, M. and Ostermann, A., Exponential integrators, Acta Numerica, 19, pp. 209–286, 2010.Google Scholar
[19] Hochbruck, M. and Ostermann, A., Explicit exponential Runge-Kuttamethods for semilinear parabolic problems, SIAM J. Numer. Anal., 43, pp. 1069–1090, 2005.Google Scholar
[20] Hochbruck, M. and Ostermann, A., Exponential multistep methods of Adams-type, BIT Numer. Math., 51, pp. 889–908, 2011.CrossRefGoogle Scholar
[21] Ju, L., Zhang, J., Zhu, L. and Du, Q., Fast explicit integration factor methods for semilinear parabolic equations, J. Sci. Comput., 62, pp. 431–455, 2015.Google Scholar
[22] Krogstad, S., Generalized integrating factor methods for stiff PDEs, J. Comput. Phys., 203, pp. 72–88, 2005.Google Scholar
[23] Kassam, A. K. and Trefethen, L. N., Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comput., 26, pp. 1214–1233, 2005.CrossRefGoogle Scholar
[24] Rebholz, L. G. and Xiao, M., On reducing the splitting error in Yosida methods for the Navier–Stokes equations with grad-div stabilization, Comput. Methods Appl. Mech. Engrg., 294, pp. 259–277, 2015.CrossRefGoogle Scholar
[25] Layton, W., Introduction to the numerical analysis of incompressible viscous flows, Society for Industrial and Applied Mathematics, 2008.Google Scholar
[26] Loan, C. V., Computational Frameworks for the Fast Fourier Transform, Society for Industrial and Applied Mathematics, 1992.Google Scholar
[27] Nie, Q., Wan, F., Zhang, Y.-T. and Liu, X., Compact integration factor methods in high spatial dimensions, J. Comput. Phys., 227, pp. 5238–5255, 2008.Google Scholar
[28] Nie, Q., Zhang, Y.-T. and Zhao, R., Efficient semi-implicit schemes for stiff systems, J. Comput. Phys., 214, pp. 521–537, 2006.Google Scholar
[29] Nochetto, R. and Pyo, J.-H., Error estimates for semi-discrete gauge methods for the Navier-Stokes equations, Math. Comput., 74, pp. 521–542, 2005.Google Scholar
[30] Nochetto, R. and Pyo, J.-H., The Gauge-Uzawa finite element method. Part I: the Navier-Stokes equations, SIAM J. Numer. Anal., 43, pp. 1043–1068, 2005.Google Scholar
[31] Pyo, J.-H. and Shen, J., Gauge-Uzawa methods for incompressible flows with variable density, J. Comput. Phys., 221, pp. 181–197, 2007.Google Scholar
[32] Temam, R., Sur l’approximation de la solution des équations de navier-stokes par la méthode des pas fractionnaires (ii), Archive for Rational Mechanics and Analysis, 33, pp. 377–385, 1969.Google Scholar
[33] Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, 1977.Google Scholar
[34] Tokman, M., Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods, J. Comput. Phys., 213, pp. 748–776, 2006.Google Scholar
[35] Wong, K. L. and Baker, A. J., A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm, Inter. J. Num. Meth. Fluids, 38(2), pp 99–123, 2002.Google Scholar
[36] Wang, C. and Liu, J.-G., Convergence of gauge method for incompressible flow, Math. Comp., 69(232), pp. 1385–1407, 2000.Google Scholar
[37] Zhu, L., Ju, L. and Zhao, W., Fast high-order compact exponential time differencing Runge–Kutta methods for second-order semilinear parabolic equations, J. Sci. Comput., 67(3), pp. 1043–1065, 2016.Google Scholar