Advertisement

Journal of Scientific Computing

, Volume 67, Issue 3, pp 1043–1065 | Cite as

Fast High-Order Compact Exponential Time Differencing Runge–Kutta Methods for Second-Order Semilinear Parabolic Equations

  • Liyong Zhu
  • Lili Ju
  • Weidong Zhao
Article

Abstract

In this paper we propose fast high-order numerical methods for solving a class of second-order semilinear parabolic equations in regular domains. The proposed methods are explicit in nature, and use exponential time differencing and Runge–Kutta approximations in combination with a linear splitting technique to achieve accurate and stable time integration. A two-step compact difference scheme is employed for spatial discretization to obtain fourth-order accuracy and make use of FFT-based fast calculations. Such methods can be applied to problems with stiff nonlinearities and boundary conditions of Dirichlet or periodic types. Linear stability analysis and various numerical experiments are also presented to demonstrate accuracy and stability of the proposed methods.

Keywords

Integrating factor Exponential time differencing Linear splitting Two-step compact difference Discrete Fourier transforms Runge–Kutta approximations 

Mathematics Subject Classification

65M06 65M22 65Y20 35K58 

References

  1. 1.
    Allen, S., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1084–1095 (1979)CrossRefGoogle Scholar
  2. 2.
    Certaine, J.: The solution of ordinary differential equations with large time constants. Mathematical methods for digital computers, pp. 128–132. Wiley, New York (1960)Google Scholar
  3. 3.
    Caplan, R.M., Carretero-Gonzalez, R.: A two-step high-order compact scheme for the Laplacian operator and its implementation in an explicit method for integrating the nonlinear schrödinger equation. J. Comput. Appl. Math. 251, 33–46 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cahn, J.W., Hillard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)CrossRefGoogle Scholar
  5. 5.
    Cox, S., Matthews, P.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Calvo, M.P., Portillo, A.M.: Variable step implementation of ETD methods for semilinear problems. Appl. Math. Comput. 196, 627–637 (2008)MathSciNetMATHGoogle Scholar
  7. 7.
    Chen, L.-Q., Shen, J.: Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Comm. 108, 147–158 (1998)CrossRefMATHGoogle Scholar
  8. 8.
    Du, Q., Gunzburger, M., Peterson, J.: Analysis and approximation of the Ginzburg–Landau model of superconductivity. SIAM Rev. 34, 54–81 (1992)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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, 450–468 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Du, Q., Zhu, W.-X.: Analysis and applications of the exponential time differencing schemes and their contour integration modifications. BIT Numer. Math. 45, 307–328 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley-Interscience, New York (1996)MATHGoogle Scholar
  12. 12.
    Hochbrucky, M., Lubich, C.: vOn Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 43, 1069–1090 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hochbruck, M., Ostermann, A.: Explicit exponential Runge–Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 19, 209–286 (2010)MathSciNetMATHGoogle Scholar
  16. 16.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations ii: Stiff and Differential Algebraic Problems. Springer, New York (1999)Google Scholar
  17. 17.
    Jiang, T., Zhang, Y.-T.: Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection-diffusion-reaction equations. J. Comput. Phys. 253, 368–388 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ju, L., Liu, X., Leng, W.: Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Dis. Cont. Dyn. Sys. B 19, 1667–1687 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ju, L., Zhang, J., Du, Q.: Fast and accurate algorithms for simulating coarsening dynamics of Cahn-Hilliard equations. Comput. Mater. Sci. 108, 272–282 (2015)CrossRefGoogle Scholar
  20. 20.
    Ju, L., Zhang, J., Zhu, L., Du, Q.: Fast Explicit Integration Factor Methods for Semilinear Parabolic Equations. J. Sci. Comput. 62, 431–455 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Krogstad, S.: Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203, 72–88 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kassam, A.K., Trefethen, L.N.: Fourth-order time stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 1214–1233 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lawson, J.: Generalized Runge–Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372–390 (1969)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Loan, C.V.: Computational Frameworks for the Fast Fourier Transform, SIAM (1992)Google Scholar
  25. 25.
    Li, B., Liu, J.-G.: Thin film epitaxy with or without slope selection. Euro. J. Appl. Math. 14, 713–743 (2003)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Nie, Q., Wan, F., Zhang, Y.-T., Liu, X.: Compact integration factor methods in high spatial dimensions. J. Comput. Phys. 227, 5238–5255 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Nie, Q., Zhang, Y.-T., Zhao, R.: Efficient semi-implicit schemes for stiff systems. J. Comput. Phys. 214, 521–537 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Pope, D.: An exponential method of numerical integration of ordinary differential equations. Comm. ACM 6, 491–493 (1963)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Qiao, Z., Sun, Z., Zhang, Z.: Stability and convergence of second-order schemes for the nonlinear epitaxial growth model without slope selection. Math. Comp. 84, 653–674 (2015)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Qiao, Z., Zhang, Z., Tang, T.: An adaptive time-stepping strategy for the molecular beam epitaxy models. SIAM J. Sci. Comput. 33, 1395–1414 (2011)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Spotz, W.F., Carey, G.F.: Extension of high-order compact schemes to time-dependent problems. Numer. Meth. PDEs 17, 657–672 (2001)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis, North-Holland (1977)Google Scholar
  33. 33.
    Whalen, P., Brio, M., Moloney, J.V.: Exponential time-differencing with embedded Runge–Kutta adaptive step control. J. Comput. Phys. 280, 579–601 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Wang, D., Zhang, L., Nie, Q.: Array-representation integration factor method for high-dimensional systems. J. Comput. Phys. 258, 585–600 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Xu, C., Tang, T.: Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44, 1759–1779 (2006)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Yang, X., Feng, J., Liu, C., Shen, J.: Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method. J. Comput. Phys. 218, 417–428 (2007)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Zhang, J., Du, Q.: Numerical studies of discrete approximations to the Allen–Cahn equation in the sharp interface limit. SIAM J. Sci. Comput. 31, 3042–3063 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematics and Systems ScienceBeihang UniversityBeijingChina
  2. 2.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  3. 3.School of MathematicsShandong UniversityJinanChina

Personalised recommendations