Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-12T06:36:40.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

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

Published online by Cambridge University Press:  11 October 2016

Liyong Zhu
Liyong Zhu*
Affiliation:
School of Mathematics and Systems Science, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191, China
*
*Corresponding author. Email:[email protected] (L. Y. Zhu)
Get access

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.

Keywords

Exponential Runge-Kutta methodexplicit schemelinear splittingdiscrete fast Fourier transformsAllen-Cahn equation

MSC classification

Secondary: 65M06: Finite difference methods 65M22: Solution of discretized equations 65Y20: Complexity and performance of numerical algorithms
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 1 , February 2017 , pp. 157 - 172
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[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. 1084–1095.CrossRefGoogle Scholar
[2] Cox, S. and Matthews, P., Exponential time differencing for stiff systems, J. Comput. Phys., 176 (2002), pp. 430–455.Google Scholar
[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. 200–209.Google Scholar
[4] Du, Q., and Zhu, W.-X., Analysis and applications of the exponential time differencing schemes, BIT Numer. Math., 45 (2005), pp. 307–328.CrossRefGoogle Scholar
[5] Hochbruck, M., Lubich, C. and Selhofer, H., Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19 (1998), pp. 1552–1574.Google Scholar
[6] Hochbruck, M. and Ostermann, A., Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43 (2005), pp. 1069–1090.Google Scholar
[7] Hochbruck, M. and Ostermann, A., Exponential integrators, Acta Numer., 19 (2010), pp. 209–286.Google Scholar
[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. 431–455.Google Scholar
[9] Krogstad, S., Generalized integrating factor methods for stiff PDEs, J. Comput. Phys., 203 (2005), pp. 72–88.Google Scholar
[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. 1591–1606.Google Scholar
[11] Luan, V. and Ostermann, A., Explicit exponential Runge-Kutta methods of high order for parabolic problems, J. Comput. Appl. Math., 256 (2014), pp. 168–179.Google Scholar
[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. 3–49.Google Scholar
[13] Maset, S. and Zennaro, M., Stability properties of explicit exponential Runge-Kutta methods, IMA J. Numer. Anal., 33 (2013), pp. 111–135.CrossRefGoogle Scholar
[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. 5238–5255.Google Scholar
[15] Wang, D., Zhang, L. and Nie, Q., Array-representation integration factor method for high-dimensional systems, J. Comput. Phys., 258 (2014), pp. 585–600.Google Scholar
[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. 417–428.Google Scholar
[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. 3042–3063.Google Scholar