Skip to main content

Advertisement

Springer Nature Link
Log in

Linearly compact scheme for 2D Sobolev equation with Burgers’ type nonlinearity

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, a bilinear three-point fourth-order compact operator is applied to solve the two-dimensional (2D) Sobolev equation with a Burgers’ type nonlinearity. In order to derive a high-order compact difference scheme, an effective reduction technique for the diffusion term is utilized to convert the original high-order evolutionary equation into a low-order system of equations equivalently, which could be discretized combining a classical linear compact operator and the bilinear compact operator. A three-level averaging technique in temporal dimension is embedded, which exports to a linearized compact difference scheme. The proposed compact difference scheme is proved to be uniquely solvable and convergent based on the energy analysis with the error estimate \(O(\tau ^{2}+{h_{1}^{4}}+{h_{2}^{4}})\) in L2-norm and H1-norm, where h1, h2, and τ denote the spatial and temporal mesh step sizes, respectively. Extensive numerical experiments are carried out to validate the theoretical results on the convergence rates even though the dynamic capillary coefficient tends to zero. Further extending the idea to the 2D Benjamin-Bona-Mahony-Burgers (BBMB) equation is available. The accuracy of the proposed scheme is also demonstrated compared with that of a second-order numerical scheme without the compact technique.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Ewing, R.E.: Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations. SIAM J. Numer. Anal. 15(6), 1125–1150 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Pany, A.K., Bajpai, S., Mishra, S.: Finite element Galerkin method for 2D Sobolev equations with Burgers’ type nonlinearity. Appl. Math. Comput. 387(15), 125113 (2020)

    MathSciNet  MATH  Google Scholar 

  3. Cao, X., Pop, I.S.: Uniqueness of weak solutions for a pseudo-parabolic equation modeling two phase flow in porous media. Appl. Math. Lett. 46, 25–30 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Shi, D.M.: On the initial boundary value problem of nonlinear the equation of the migration of the moisture in soil. Acta. Math. Appl. Sin. 13(1), 31–38 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Böhm, M., Showalter, R.E.: A nonlinear pseudoparabolic diffusion equation. SIAM J. Math. Anal. 16, 980–999 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barenblett, G.I., Zheltov, I.P., Kochian, I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24(5), 1286–1303 (1990)

    Article  Google Scholar 

  7. Cao, X., Pop, I.S.: Degenerate two-phase porous media flow model with dynamic capillarity. J. Differ. Equ. 260(3), 2418–2456 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ting, T.W.: A cooling process according to two-temperature theory of heat conduction. J. Math. Anal. Appl. 45(1), 23–31 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Showalter, R.E.: Existence and representation theorems for a semilinear Sobolev equation in banach space. SIAM J. Math. Anal. 3(3), 527–543 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mikelić, A.: A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure. J. Differ. Equ. 248(6), 1561–1577 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cuesta, C., Hulshof, J.: A model problem for groundwater flow with dynamic capillary pressure: stability of travelling waves. Nonlinear Anal. TMA 52(4), 1199–1218 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bertsch, M., Smarrazzo, F., Tesei, A.: Pseudoparabolic regularization of forward-backward parabolic equations: a logarithmic nonlinearity. Anal. PDE 6, 1719–1754 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cao, X., Pop, I.S.: Two-phase porous media flows with dynamic capillary effects and hysteresis: uniqueness of weak solutions. Comput. Math. Appl. 69, 688–695 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan, Y., Pop, I.S.: A class of degenerate pseudo-parabolic equations: existence, uniqueness of weak solutions, and error estimates for the Euler-implicit discretization. Math. Methods Appl. Sci. 34, 2329–2339 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Koch, J., Rätz, A., Schweizer, B.: Two-phase flow equations with a dynamic capillary pressure. Eur. J. Appl. Math. 24(1), 49–75 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Milisic, J.P.: The unsaturated flow in porous media with dynamic capillary pressure. J. Differ. Equ. 264, 5629–5658 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Seam, N., Vallet, G.: Existence results for nonlinear pseudoparabolic problems. Nonlinear Anal. Real World Appl. 12, 2625–2639 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cao, X., Mitra, K.: Error estimates for a mixed finite element discretization of a two-phase porous media flow model with dynamic capillarity. J. Comput. Appl. Math. 252, 164–178 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cao, X., Nemadjieu, S.F., Pop, I.S.: Convergence of an MPFA finite volume scheme for a two-phase porous media flow model with dynamic capillarity. IMA J. Numer. Anal. 39, 512–544 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Ewing, R.E.: Numerical solution of Sobolev partial differential equations. SIAM J. Numer. Anal. 12(3), 345–363 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nakao, M.T.: Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension. Numer. Math. 47, 139–157 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lin, Y.P., Zhang, T.: Finite element methods for nonlinear Sobolev equations with nonlinear boundary conditions. J. Math. Anal. Appl. 165, 180–191 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhao, Z., Li, H., Luo, Z.: Analysis of a space-time continuous Galerkin method for convection-dominated Sobolev equations. Comput. Math. Appl. 73, 1643–1656 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhao, Z., Li, H., Wang, J.: The study of a continuous Galerkin method for Sobolev equation with space-time variable coefficients. Appl. Math. Comput. 401, 126021 (2021)

    MathSciNet  MATH  Google Scholar 

  25. Karpinski, S., Pop, I.S.: Analysis of an interior penalty discontinuous Galerkin scheme for two phase flow in porous media with dynamic capillarity effects. Numer. Math. 136, 249–286 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Abreu, E., Duran, A.: Spectral discretizations analysis with time strong stability preserving properties for pseudo-parabolic models. Comput. Math. Appl. 102, 15–44 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ford, W.H., Ting, T.W.: Uniform error estimates for difference approximations to nonlinear pseudo-parabolic partial differential equations. SIAM J. Numer. Anal. 11, 155–169 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  28. Helmig, R., Weiss, A., Wohlmuth, B.: Dynamic capillary effects in heterogeneous porous media. Comput. Geosci. 11, 261–274 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lunowa, S.B., Pop, I.S., Koren, B.: Linearized domain decomposition methods for two-phase porous media flow models involving dynamic capillarity and hysteresis. Comput. Methods Appl. Mech. Eng. 372, 113364 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Peszynska, M., Yi, S.-Y.: Numerical methods for unsaturated flow with dynamic capillary pressure in heterogeneous porous media. Int. J. Numer. Anal. Model. 5, 126–149 (2008)

    MathSciNet  MATH  Google Scholar 

  31. Abreu, E., Ferraz, P., Vieira, J.: Numerical resolution of a pseudo-parabolic Buckley-Leverett model with gravity and dynamic capillary pressure in heterogeneous porous media. J. Comput. Phys. 411, 109395 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, C., Tang, C.: One-parameter orthogonal spline collocation methods for nonlinear two-dimensional Sobolev equations with time-variable delay. Commun. Nonlinear Sci. Numer. Simul. 108, 106233 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  33. Holden, H., Lubich, C., Risebro, N.H.: Operator splitting for partial differential equations with Burgers nonlinearity. Math. Comput. 82 (281), 173–185 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Cuesta, C.M., Pop, I.S.: Numerical schemes for a pseudo-parabolic Burgers equation: discontinuous data and long-time behaviour. J. Comput. Appl. Math. 224, 269–283 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yang, H.: Superconvergence error estimate of Galerkin method for Sobolev equation with Burgers’ type nonlinearity. Appl. Numer. Math. 168, 13–22 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kundu, S., Pani, A.K.: Global stabilization of two dimensional viscous Burgers’ equation by nonlinear neumann boundary feedback control and its finite element analysis. J. Sci. Comput. 84, Article 45 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pany, A.K., Kundu, S.: Optimal error estimates for semidiscrete Galerkin approximations to multi-dimensional Sobolev equations with Burgers’ type nonlinearity. In: Proceeding of the International Conference on Numerical Analysis and Optimization (NAO 2017), SQU, Muscat, pp 209–227. Springer (2018)

  38. Mishra, S., Pany, A.K.: Completely discrete schemes for 2D Sobolev equations with Burgers’ type nonlinearity. Numer. Algor. https://doi.org/10.1007/s11075-021-01218-2 (2021)

  39. Chu, P., Fan, C.: A three-point combined compact difference scheme. J. Comput. Phys. 140, 370–399 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, X., Zhang, Q., Sun, Z.: The pointwise error estimates of two energy-preserving fourth-order compact schemes for viscous Burgers’ equation. Adv. Comput. Math. 47, Article 23 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sun, Z.: The Numerical Methods for Partial Differential Equations (3rd Edition). Science Press, Beijing (2022). (in Chinese)

    Google Scholar 

  42. Numerov, B.: Note on the numerical integration of d2x/dt2 = f(x, t). Astron. Notes 230, 359–364 (1927)

    MATH  Google Scholar 

  43. Kuo, P.Y., Sanz-Serna, J.M.: Convergence of methods for the numerical solution of the Korteweg-de Vries equation. IMA J. Numer. Anal. 1(2), 215–221 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhang, Q., Liu, L.: Convergence and stability in maximum norms of linearized fourth-order conservative compact scheme for Benjamin-Bona-Mahony-Burgers’ equation. J. Sci. Comput. 87, Article 59 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sun, Z.: Finite Difference Methods for Nonlinear Evolution Equations. Science Press, Beijing (2018)

    Google Scholar 

  46. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate. J. Comput. Appl. Math. 286, 211–231 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  47. Haq, S., Ghafoor, A., Hussain, M., Arifeen, S.: Numerical solutions of two dimensional Sobolev and generalized Benjamin-Bona-Mahony-Burgers equations via Haar wavelets. Comput. Math. Appl. 77, 565–575 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees for their helpful comments and suggestion for the improvement of the paper.

Funding

This work was supported in part by projects funded by the Natural Science Foundation of China (Grant No. 11501514).

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Qifeng Zhang & Yifan Qin

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Zhi-zhong Sun

Authors
  1. Qifeng Zhang
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Yifan Qin
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Zhi-zhong Sun
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Qifeng Zhang.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Data availability

All data or codes generated or used during the study are available from the corresponding author by request.

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

Zhang, Q., Qin, Y. & Sun, Zz. Linearly compact scheme for 2D Sobolev equation with Burgers’ type nonlinearity. Numer Algor 91, 1081–1114 (2022). https://doi.org/10.1007/s11075-022-01293-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-022-01293-z

Keywords

Mathematics Subject Classification (2010)