Skip to main content
SpringerLink
Log in

Iterative acceleration methods with second-order time accuracy for nonlinear diffusion equations

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

For a two-layer finite difference scheme with second-order time accuracy of nonlinear diffusion equations, we present three iterative solving algorithms, including Picard, Picard-Newton and derivative-free Picard-Newton iterations. The main purpose of this paper is to solve the two-layer scheme efficiently and accurately, and give strict theoretical proofs of the convergence and efficiency of the three iterative methods. For the three iterative methods, two strategies are adopted to offer iterative initial evaluations, one is to use the value at the previous time step with first-order accuracy, the other is an extrapolation with second-order accuracy. With an induction reasoning technique, we prove that the solutions of the three iterative methods all converge to the exact solution of the diffusion problem with second-order accuracy both in space and time after two nonlinear iteration steps, even if the first-order initial value is used. It is also proved that the solutions of Picard iteration converge linearly to the solution of the discrete scheme, while Picard-Newton and derivative-free Picard-Newton iterative solutions converge with a quadratic speed. Moreover, no difference occurs in convergent speed with the two different initial values. Finally, some numerical tests are presented to verify our theoretical results, which show that compared with Picard iteration, Picard-Newton and derivative-free Picard-Newton iterations are more efficient for solving nonlinear problems.

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

Similar content being viewed by others

References

  1. Akrivis, G., Makridakis, C., Nochetto, R.: A posteriori error estimates for the Crank-Nicolson method for parabolic equations. Math. Comput. 75, 511–531 (2006)

    Article  MathSciNet  Google Scholar 

  2. Chen, G., Chacón, L., Leibs, C.A., Knoll, D.A., Taitano, W.: Fluid preconditioning for Newton-Krylov-based, fully implicit, electrostatic particle-in-cell simulations. J. Comput. Phys. 258, 555–567 (2014)

    Article  MathSciNet  Google Scholar 

  3. Cui, X., Yuan, G.W., Shen, Z.J.: An efficient solving method for nonlinear convection diffusion equation. Int. J. Numer Meth. Heat Fluid Flow. 28, 173–187 (2018)

    Article  Google Scholar 

  4. Cui, X., Yuan, G.W., Yue, J.Y.: Numerical analysis and iteration acceleration of a fully implicit scheme for nonlinear diffusion problem with second-order time evolution. Numer. Meth. Part. D. E. 32, 121–140 (2016)

    Article  MathSciNet  Google Scholar 

  5. Cui, X., Yuan, G.W., Zhao, F.: Analysis on a numerical scheme with second-order time accuracy for nonlinear diffusion equations. Submitted (2019)

  6. Cui, X., Yue, J.Y.: Property analysis and quick solutions for nonlinear discrete schemes for conservative diffusion equation. Math. Numer. Sin. 37, 227–246 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Cui, X., Yue, J.Y., Yuan, G.W.: Nonlinear scheme with high accuracy for nonlinear coupled parabolic-hyperbolic system. J. Comput. Appl. Math. 235, 3527–3540 (2011)

    Article  MathSciNet  Google Scholar 

  8. Dai, W.L., Woodward, P.R.: Numerical simulations for nonlinear heat transfer in a system of multimaterials. J. Comput. Phys. 139(1), 58–78 (1998)

    Article  Google Scholar 

  9. Dai, W.W., Scannapieco, A.J.: Interface- and discontinuity-aware numerical schemes for plasma 3-T radiation diffusion in two and three dimensions. J. Comput. Phys. 300, 643–664 (2015)

    Article  MathSciNet  Google Scholar 

  10. Dai, W.W., Scannapieco, A.J.: Second-order accurate interface- and discontinuity-aware diffusion solvers in two and three dimensions. J. Comput. Phys. 281, 982–1002 (2015)

    Article  MathSciNet  Google Scholar 

  11. Douglas, J., Dupont, T., Ewing, R.E.: Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem. SIAM J. Numer. Anal. 16 (3), 503–522 (1979)

    Article  MathSciNet  Google Scholar 

  12. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. Techniques of Scientific Computing, Part III, Handbook of Numerical Analysis, VII 7(4), 713–1020 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Kadioglu, S.Y., Knoll, D.A.: A Jacobian-free Newton-Krylov implicit-explicit time integration method for incompressible flow problems. Commun. Comput. Phys. 13, 1408–1431 (2013)

    Article  MathSciNet  Google Scholar 

  14. Karaa, S., Zhang, J.: Analysis of stationary iterative methods for the discrete convection-diffusion equation with a 9-point compact scheme. J. Comput. Appl. Math. 154, 447–476 (2003)

    Article  MathSciNet  Google Scholar 

  15. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: A survey of approaches and applications. J. Comput. Phys. 193, 357–397 (2004)

    Article  MathSciNet  Google Scholar 

  16. Knoll, D.A., Park, H., Smith, K.: Application of the Jacobian-free Newton-Krylov method to nonlinear acceleration of transport source iteration in slab geometry. Nucl. Sci. Eng. 167, 122–132 (2011)

    Article  Google Scholar 

  17. Li, B.Y., Gao, H.D., Sun, W.W.: Unconditionally optimal error estimates of a Crank-Nicolson Galerkin method for the nonlinear thermistor equations. SIAM J. Numer. Anal. 52, 933–954 (2014)

    Article  MathSciNet  Google Scholar 

  18. Li, B.Y., Sun, W.W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51(4), 1959–1977 (2012)

    Article  MathSciNet  Google Scholar 

  19. Luo, Z.D.: A stabilized Crank-Nicolson mixed finite volume element formulation for the non-stationary incompressible Boussinesq equations. J. Sci. Comput. 66, 555–576 (2016)

    Article  MathSciNet  Google Scholar 

  20. Olson, G.L.: Efficient solution of multi-dimensional flux-limited nonequilibrium radiation diffusion coupled to material conduction with second-order time discretization. J. Comput. Phys. 226, 1181–1195 (2007)

    Article  Google Scholar 

  21. Rauenzahn, R.M., Mousseau, V.A., Knoll, D.A.: Temporal accuracy of the nonequilibrium radiation diffusion equations employing a Saha ionization model. Comput. Phys. Commun. 172, 109–118 (2005)

    Article  Google Scholar 

  22. Schneider, M., Flemisch, B., Helmig, R.: Monotone nonlinear finite-volume method for nonisothermal two-phase two-component flow in porous media. Int. J. Numer. Meth. Fl. 84, 352–381 (2017)

    Article  MathSciNet  Google Scholar 

  23. Sun, W.W., Sun, Z.Z.: Finite difference methods for a nonlinear and strongly coupled heat and moisture transport system in textile materials. Numer. Math. 120, 153–187 (2012)

    Article  MathSciNet  Google Scholar 

  24. Wang, T.C., Zhao, X.F., Jiang, J.P.: Unconditional and optimal H2-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrodinger̈ equation in high dimensions. Adv. Comput. Math. 44, 477–503 (2018)

    Article  MathSciNet  Google Scholar 

  25. Yuan, G.W., Hang, X.D.: Acceleration methods of nonlinear iteration for nonlinear parabolic equations. J. Comput. Math. 24, 412–424 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Yuan, G.W., Hang, X.D., Sheng, Z.Q., Yue, J.Y.: Progress in numerical methods for radiation diffusion equations. Chinese J. Comput. Phys. 26, 475–500 (2009)

    Google Scholar 

  27. Yuan, G.W., Sheng, Z.Q., Hang, XD., Yao, Y.Z., Chang, L.N., Yue, J.Y.: Computational Methods for Diffusion Equation. Science Press, Beijing (2015)

    Google Scholar 

  28. Yue, J.Y., Yuan, G.W.: Picard-Newton iterative method with time step control for multimaterial non-equilibrium radiation diffusion problem. Commun. Comput. Phys. 10, 844–866 (2011)

    Article  MathSciNet  Google Scholar 

  29. Zhang, G.D., Yang, J.J., Bi, C.J.: Second order unconditionally convergent and energy stable linearized scheme for MHD equations. Adv. Comput. Math. 44, 505–540 (2018)

    Article  MathSciNet  Google Scholar 

  30. Zhou, Y.L.: Applications of Discrete Functional Analysis to the Finite Difference Method. International Academic Publishers, Beijing (1990)

    Google Scholar 

Download references

Funding

This work is financially supported by the National Natural Science Foundation of China (11871112, 11571048, U1630249, 11971069), the Science Challenge Project (TZ2016002), Yu Min Foundation, and the Foundation of LCP.

Author information

Authors and Affiliations

  1. Graduate School of China Academy of Engineering Physics, Beijing, 100088, China

    Fei Zhao

  2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-26, Beijing, 100088, China

    Xia Cui & Guangwei Yuan

Authors
  1. Fei Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xia Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guangwei Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangwei Yuan.

Additional information

Communicated by: Aihui Zhou

Publisher’s note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, F., Cui, X. & Yuan, G. Iterative acceleration methods with second-order time accuracy for nonlinear diffusion equations. Adv Comput Math 46, 7 (2020). https://doi.org/10.1007/s10444-020-09756-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-020-09756-4

Keywords

Mathematics subject classification (2010)

Search

Navigation