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Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

  • Liyong Zhu (a1)

Abstract

In this paper we develop explicit fast exponential Runge-Kutta methods for the numerical solutions of a class of parabolic equations. By incorporating the linear splitting technique into the explicit exponential Runge-Kutta schemes, we are able to greatly improve the numerical stability. The proposed numerical methods could be fast implemented through use of decompositions of compact spatial difference operators on a regular mesh together with discrete fast Fourier transform techniques. The exponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximations with variable time step sizes, as well as applied to stiff nonlinearity and boundary conditions of different types. Linear stabilities of the proposed schemes and their comparison with other schemes are presented. We also numerically demonstrate accuracy, stability and robustness of the proposed method through some typical model problems.

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Corresponding author

*Corresponding author. Email: liyongzhu@buaa.edu.cn (L. Y. Zhu)

References

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[1] Allen, S. and Cahn, J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), pp. 10841095.
[2] Cox, S. and Matthews, P., Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), pp. 430455.
[3] Du, Q., and Zhu, W.-X., Stability analysis and applications of the exponential time differencing schemes and their contour integration modifications, J. Comput. Math., 22 (2004), pp. 200209.
[4] Du, Q., and Zhu, W.-X., Analysis and applications of the exponential time differencing schemes, BIT Numer. Math., 45 (2005), pp. 307328.
[5] Hochbruck, M., Lubich, C. and Selhofer, H., Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19 (1998), pp. 15521574.
[6] Hochbruck, M. and Ostermann, A., Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43 (2005), pp. 10691090.
[7] Hochbruck, M. and Ostermann, A., Exponential integrators, Acta Numer., 19 (2010), pp. 209286.
[8] Ju, L., Zhang, J., Zhu, L. and Du, Q., Fast explicit integration factor methods for semilinear parabolic equations, J. Sci. Comput., 62 (2015), pp. 431455.
[9] Krogstad, S., Generalized integrating factor methods for stiff PDEs, J. Comput. Phys., 203 (2005), pp. 7288.
[10] Li, Y., Lee, H.-G., Jeong, D., and Kim, J., An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation, Comput. Math. Appl., 60 (2010), pp. 15911606.
[11] Luan, V. and Ostermann, A., Explicit exponential Runge-Kutta methods of high order for parabolic problems, J. Comput. Appl. Math., 256 (2014), pp. 168179.
[12] Moler, C. and Loan, C. V., Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45 (2003), pp. 349.
[13] Maset, S. and Zennaro, M., Stability properties of explicit exponential Runge-Kutta methods, IMA J. Numer. Anal., 33 (2013), pp. 111135.
[14] Nie, Q., Wan, F., Zhang, Y.-T. and Liu, X.-F., Compact integration factor methods in high spatial dimensions, J. Comput. Phys., 227 (2008), pp. 52385255.
[15] Wang, D., Zhang, L. and Nie, Q., Array-representation integration factor method for high-dimensional systems, J. Comput. Phys., 258 (2014), pp. 585600.
[16] Yang, X., Feng, J., Liu, C. and Shen, J., Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, J. Comput. Phys., 218 (2007), pp. 417428.
[17] Zhang, J. and Du, Q., Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM J. Sci. Comput., 31 (2009), pp. 30423063.

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Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

  • Liyong Zhu (a1)

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