Fast Explicit Integration Factor Methods for Semilinear Parabolic Equations

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• 作者：Lili Ju ; Jian Zhang ; Liyong Zhu ; Qiang Du
• 关键词：Integration factor method ; Explicit scheme ; Multistep approximation ; Discrete fast transforms ; Diffusion–reaction equation ; Allen–Cahn equation ; 65M06 ; 65M22 ; 65Y20
• 刊名：Journal of Scientific Computing
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：62
• 期：2
• 页码：431-455
• 全文大小：837 KB
• 参考文献：1. Allen, S., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1084-095 (1979) CrossRef
  2. Chen, L.-Q., Shen, J.: Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun. 108, 147-58 (1998) CrossRef
  3. Cox, S., Matthews, P.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430-55 (2002) CrossRef
  4. Du, Q., Liu, C., Wang, X.: A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys. 198, 450-68 (2004) CrossRef
  5. Du, Q., Zhu, W.-X.: Stability analysis and applications of the exponential time differencing schemes and their contour integration modifications. J. Comput. Math. 22, 200-09 (2004)
  6. Du, Q., Zhu, W.-X.: Analysis and applications of the exponential time differencing schemes. BIT Numer. Math. 45, 307-28 (2005) CrossRef
  7. Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33, 635-81 (1991)
  8. Feng, X., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33-5 (2003) CrossRef
  9. Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley-Interscience, New York (1996)
  10. Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552-574 (1998) CrossRef
  11. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209-86 (2010) CrossRef
  12. Kessler, D., Nochetto, R.H., Schmidt, A.: A posteriori error control for the Allen–Cahn problem: circumventing Gronwall’s inequality. Math. Model. Numer. Anal. 38, 129-42 (2004) CrossRef
  13. Krogstad, S.: Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203, 72-8 (2005) CrossRef
  14. Loan, C.V.: Computational Frameworks for the Fast Fourier Transform. SIAM, Philadelphia (1992) CrossRef
  15. Li, Y., Lee, H.-G., Jeong, D., Kim, J.: An unconditionally stable hybrid numerical method for solving the Allen–Cahn equation. Comput. Math. Appl. 60, 1591-606 (2010) CrossRef
  16. de Mottoni, P., Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Am. Math. Soc. 347, 1533-589 (1989) CrossRef
  17. Nie, Q., Wan, F., Zhang, Y.-T., Liu, X.-F.: Compact integration factor methods in high spatial dimensions. J. Comput. Phys. 227, 5238-255 (2008) CrossRef
  18. Nie, Q., Zhang, Y.-T., Zhao, R.: Efficient semi-implicit schemes for stiff systems. J. Comput. Phys. 214, 52
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691

文摘

In this paper, an explicit numerical method and its fast implementation are proposed and discussed for the solution of a wide class of semilinear parabolic equations including the Allen–Cahn equation as a special case. The method combines decompositions of compact spatial difference operators on a regular mesh with stable and accurate exponential time integrators and efficient discrete FFT-based algorithms. It can deal with stiff nonlinearity and both homogeneous and inhomogeneous boundary conditions of different types based on multistep approximations and analytic evaluations of time integrals. Numerical experiments demonstrate effectiveness of the new method for both linear and nonlinear model problems.