Skip to main content
SpringerLink
Log in

Overlapping domain decomposition based exponential time differencing methods for semilinear parabolic equations

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen–Cahn equation as a special case. We first study the semi-discrete system under the standard central difference spatial discretization and prove the equivalence between the monodomain problem and the corresponding multidomain problem obtained by the Schwarz waveform relaxation iteration. Then we develop the fully discrete localized exponential time differencing schemes and, by establishing the maximum bound principle, prove the convergence of the fully discrete localized solutions to the exact semi-discrete solution and the convergence of the iterative solutions. Numerical experiments are carried out to verify the theoretical results in one-dimensional space and test the convergence and accuracy of the proposed algorithms with different numbers of subdomains in two-dimensional space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33, 488–511 (2011)

    Article  MathSciNet  Google Scholar 

  2. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)

    Article  Google Scholar 

  3. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

    Article  Google Scholar 

  4. Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)

    Article  MathSciNet  Google Scholar 

  5. Du, Q., Gunzburger, M.D., Peterson, J.S.: Analysis and approximation of the Ginzburg–Landau model of superconductivity. SIAM Rev. 34, 54–81 (1992)

    Article  MathSciNet  Google Scholar 

  6. Du, Q., Ju, L., Li, X., Qiao, Z.: Maximum principle preserving exponential time differencing schemes for the nonlocal Allen–Cahn equation. SIAM J. Numer. Anal. 57, 875–898 (2019)

    Article  MathSciNet  Google Scholar 

  7. Du, Q., Ju, L., Li, X., Qiao, Z.: Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes. SIAM Rev. (2020, to appear)

  8. Elder, K.R., Katakowski, M., Haataja, M., Grant, M.: Modeling elasticity in crystal growth. Phys. Rev. Lett. 88, 245–701 (2002)

    Article  Google Scholar 

  9. Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45, 1097–1123 (1992)

    Article  MathSciNet  Google Scholar 

  10. Gander, M.J.: A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations. Numer. Linear Algebra Appl. 6, 125–145 (1999)

    Article  MathSciNet  Google Scholar 

  11. Gander, M.J., Stuart, A.M.: Space-time continuous analysis of waveform relaxation for the heat equation. SIAM J. Sci. Comput. 19, 2014–2031 (1998)

    Article  MathSciNet  Google Scholar 

  12. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MathSciNet  Google Scholar 

  13. Gustafsson, B., Kreiss, H.O., Oliger, J.: Time-Dependent Problems and Difference Methods, 2nd edn. Wiley, Hoboken (2013)

    Book  Google Scholar 

  14. Hoang, T.T.P., Ju, L., Wang, Z.: Overlapping localized exponential time differencing methods for diffusion problems. Commum. Math. Sci. 16, 1531–1555 (2018)

    Article  MathSciNet  Google Scholar 

  15. Hoang, T.T.P., Ju, L., Wang, Z.: Nonoverlapping localized exponential time differencing methods for diffusion problems. J. Sci. Comput. 82, 37 (2020)

    Article  MathSciNet  Google Scholar 

  16. Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)

    Article  MathSciNet  Google Scholar 

  17. Hochbruck, M., Ostermann, A.: Explicit exponential Runge—Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43, 1069–1090 (2005)

    Article  MathSciNet  Google Scholar 

  18. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  Google Scholar 

  19. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  20. Hu, G.Y., O’Connell, R.F.: Analytical inversion of symmetric tridiagonal matrices. J. Phys. A 29, 1511–1513 (1996)

    Article  MathSciNet  Google Scholar 

  21. Isherwood, L., Grant, Z.J., Gottlieb, S.: Strong stability preserving integrating factor Runge-Kutta methods. SIAM J. Numer. Anal. 56, 3276–3307 (2018)

    Article  MathSciNet  Google Scholar 

  22. Ju, L., Li, X., Qiao, Z., Zhang, H.: Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection. Math. Comput. 87, 1859–1885 (2018)

    Article  MathSciNet  Google Scholar 

  23. 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)

    Article  Google Scholar 

  24. Ju, L., Zhang, J., Zhu, L., Du, Q.: Fast explicit integration factor methods for semilinear parabolic equations. J. Sci. Comput. 62, 431–455 (2015)

    Article  MathSciNet  Google Scholar 

  25. Kleefeld, B., Martín-Vaquero, J.: SERK2v2: a new second-order stabilized explicit Runge–Kutta method for stiff problems. Numer. Methods Part. Differ. Equ. 29, 170–185 (2013)

    Article  MathSciNet  Google Scholar 

  26. Lawson, J.D.: Generalized Runge–Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372–380 (1967)

    Article  MathSciNet  Google Scholar 

  27. Lazer, A.C.: Characteristic exp nents and diagonally dominant linear differential systems. J. Math. Anal. Appl. 35, 215–229 (1971)

    Article  MathSciNet  Google Scholar 

  28. Li, B., Liu, J.G.: Thin film epitaxy with or without slope selection. Eur. J. Appl. Math. 14, 713–743 (2003)

    Article  MathSciNet  Google Scholar 

  29. Medovikov, A.A.: High order explicit methods for parabolic equations. BIT 38, 372–390 (1998)

    Article  MathSciNet  Google Scholar 

  30. Shu, C.W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  Google Scholar 

  31. Temam, R.: Navier–Stokes E quations: Theory and Numerical Analysis. North-Holland Publishing Co., New York (1977)

    Google Scholar 

  32. Zhang, J., Zhou, Z., Wang, Y., Ju, L., Du, Q., Chi, X., Xu, D., Chen, D., Liu, Y., Liu, Z.: Extreme-scale phase field simulations of coarsening dynamics on the Sunway Taihulight supercomputer. In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (SC’16). IEEE Press, Piscataway (2016)

  33. Zhu, L., Ju, L., Zhao, W.: Fast high-order compact exponential time differencing Runge–Kutta methods for second-order semilinear parabolic equations. J. Sci. Comput. 67, 1043–1065 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. Xiao Li

    Present address: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Xiao Li & Lili Ju

  2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    Thi-Thao-Phuong Hoang

Authors
  1. Xiao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lili Ju
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thi-Thao-Phuong Hoang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao Li.

Additional information

Communicated by Christian Lubich.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

X. Li’s work is partially supported by National Natural Science Foundation of China grant 11801024. L. Ju’s work is partially supported by US National Science Foundation grant DMS-1818438 and US Department of Energy grants DE-SC0016540 and DE-SC0020270.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X., Ju, L. & Hoang, TTP. Overlapping domain decomposition based exponential time differencing methods for semilinear parabolic equations. Bit Numer Math 61, 1–36 (2021). https://doi.org/10.1007/s10543-020-00817-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-020-00817-0

Keywords

Mathematics Subject Classification

Search

Navigation