Skip to main content
SpringerLink
Log in

An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a \(C^0\) finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.

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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Abdulloev, K.O., Bogolubsky, I.L., Makhankov, V.G.: One more example of inelastic soliton interaction. Phys. Lett. A 56, 427–428 (1976)

    Article  MathSciNet  Google Scholar 

  2. Avrin, J., Goldstein, J.A.: Global existence for the Benjamin–Bona–Mahony equation in arbitrary dimensions. Nonlinear Anal. 9, 861–865 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baines, M.J.: Moving Finite Elements. Oxford University Press, Oxford (1994)

    MATH  Google Scholar 

  4. Baines, M.J., Hubbard, M.E., Jimack, P.K.: Velocity-based moving mesh methods for nonlinear partial differential equations. Commun. Comput. Phys. 10, 509–576 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R Soc. Lond. A 227, 47–78 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bona, J.L., McKinney, W.R., Restrepo, J.M.: Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation. J. Nonlinear Sci. 10, 603–638 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bona, J.L., Pritchard, W.G., Scott, L.R.: An evaluation of a model equation for water waves. Philos. Trans. R. Soc. Lond. A 302, 457–510 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. Budd, C.J., Huang, W., Russell, R.D.: Adaptivity with moving grids. Acta Numer. 18, 111–241 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Calogero, F., Degasperis, A.: Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations. North-Holland, New York (1982)

    MATH  Google Scholar 

  10. Calvert, B.: The equation \(A(t, u(t))^{\prime }+B(t, u(t))=0\). Math. Proc. Camb. Philos. Soc. 79, 545–561 (1976)

    Article  MATH  Google Scholar 

  11. Daǧ, İ., Saka, B., Irk, D.: Application of cubic B-splines for numerical solution of the RLW equation. Appl. Math. Comput. 159, 373–389 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Dehghan, M., Salehi, R.: The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas. Comput. Phys. Commun. 182, 2540–2549 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dogan, A.: Numerical solution of RLW equation using linear finite elements within Galerkin’s method. Appl. Math. Model. 26, 771–783 (2002)

    Article  MATH  Google Scholar 

  14. Eilbeck, J.C., McGuire, G.R.: Numerical study of the regularized long-wave equation I: numerical methods. J. Comput. Phys. 19, 43–57 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eilbeck, J.C., McGuire, G.R.: Numerical study of the regularized long-wave equation II: Iinteraction of solitary waves. J. Comput. Phys. 23, 63–73 (1975)

    Article  MATH  Google Scholar 

  16. Gao, F., Qiu, J., Zhang, Q.: Local discontinuous Galerkin finite element method and error estimates for one class of Sobolev equation. J. Sci. Comput. 41, 436–460 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gao, Y., Mei, L.: Mixed Galerkin finite element methods for modified regularized long wave equation. Appl. Math. Comput. 258, 267–281 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Goldstein, J.A., Wichnoski, B.J.: On the Benjamin–Bona–Mahony equation in higher dimensions. Nonlinear Anal. 4, 665–675 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. González-Pinto, S., Montijano, J.I., Pérez-Rodríguez, S.: Two-step error estimators for implicit Runge–Kutta methods applied to stiff systems. ACM Trans. Math. Softw. 30(1), 1–18 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gu, H., Chen, N.: Least-squares mixed finite element methods for the RLW equations. Numer. Meth. Partial Differ. Equ. 24, 749–758 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo, B.-Y., Cao, W.-M.: The fourier pseudospectral method with a restrain operator for the RLW equation. J. Comput. Phys. 74, 110–126 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II, Volume 14 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (1996). Stiff and differential-algebraic problems

  23. Huang, W.: Variational mesh adaptation: isotropy and equidistribution. J. Comput. Phys. 174, 903–924 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Huang, W.: Mathematical principles of anisotropic mesh adaptation. Commun. Comput. Phys. 1, 276–310 (2006)

    MATH  Google Scholar 

  25. Huang, W., Kamenski, L.: A geometric discretization and a simple implementation for variational mesh generation and adaptation. J. Comput. Phys. 301, 322–337 (2015). (arXiv:1410.7872)

    Article  MathSciNet  MATH  Google Scholar 

  26. Huang, W., Kamenski, L.: On the mesh nonsingularity of the moving mesh PDE method. submitted (2015). arXiv:1512.04971

  27. Huang, W., Ren, Y., Russell, R.D.: Moving mesh methods based on moving mesh partial differential equations. J. Comput. Phys. 113, 279–290 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Huang, W., Ren, Y., Russell, R.D.: Moving mesh partial differential equations (MMPDEs) based upon the equidistribution principle. SIAM J. Numer. Anal. 31, 709–730 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Huang, W., Russell, R.D.: Adaptive Moving Mesh Methods. Springer, New York (2011). Applied Mathematical Sciences Series, Vol. 174

  30. Huang, W., Sun, W.: Variational mesh adaptation II: error estimates and monitor functions. J. Comput. Phys. 184, 619–648 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jimack, P.K., Wathen, A.J.: Temporal derivatives in the finite-element method on continuously deforming grids. SIAM J. Numer. Anal. 28, 990–1003 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kamenski, L., Huang, W.: How a nonconvergent recovered Hessian works in mesh adaptation. SIAM J. Numer. Anal. 52, 1692–1708 (2014). (arXiv:1211.2877)

    Article  MathSciNet  MATH  Google Scholar 

  33. Luo, Z., Liu, R.: Mixed finite element analysis and numerical solitary solution for the RLW equation. SIAM J. Numer. Anal. 36, 89–104 (1999). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ma, W.X., Fuchssteiner, B.: Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation. Int. J. Non-Linear Mech. 31, 329–338 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  36. Medeiros, L.A., Miranda, M.M.: Weak solutions for a nonlinear dispersive equation. J. Math. Anal. Appl. 59, 432–441 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mei, L., Chen, Y.: Explicit multistep method for the numerical solution of RLW equation. Appl. Math. Comput. 218, 9547–9554 (2012)

    MathSciNet  MATH  Google Scholar 

  38. Mei, L., Chen, Y.: Numerical solutions of RLW equation using Galerkin method with extrapolation techniques. Comput. Phys. Commun. 183, 1609–1616 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mei, L., Gao, Y., Chen, Z.: Numerical study using explicit multistep Galerkin finite element method for the MRLW equation. Numer. Meth. Partial Differ. Equ. 31, 1875–1889 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Olver, P.J.: Euler operators and conservation laws of the BBM equation. Math. Proc. Camb. Philos. Soc. 85, 143–160 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  41. Peregrine, D.H.: Calculations of the development of an undular bore. J. Fluid Mech. 25(2), 321–330 (1966)

    Article  MathSciNet  Google Scholar 

  42. Rosenau, P.: A quasi-continuous description of a nonlinear transmission line. Phys. Scr. 34, 827–829 (1986)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Siraj-ul-Islam, S., Haq, Ali, A.: A meshfree method for the numerical solution of the RLW equation. J. Comput. Appl. Math. 223, 997–1012 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Spayd, K., Shearer, M.: The Buckley-Leverett equation with dynamic capillary pressure. SIAM J. Appl. Math. 71, 1088–1108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Tang, T.: Moving mesh methods for computational fluid dynamics flow and transport. In: Recent Advances in Adaptive Computation (Hangzhou, 2004), volume 383 of AMS Contemporary Mathematics, pp. 141–173. Amer. Math. Soc, Providence, RI (2005)

  47. Tian, B., Li, W., Gao, Y.-T.: On the two-dimensional regularized long-wave equation in fluids and plasmas. Acta Mech. 160, 235–239 (2003)

    Article  MATH  Google Scholar 

  48. Yang, H., Xu, Z., Yang, D., Feng, X., Yin B., Dong, H.: ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect. Adv. Differ. Equ. 167, 1–22 (2016)

  49. Zaki, S.: Solitary waves of the split RLW equation. Comput. Phys. Commun. 138, 80–91 (2001)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The work was partially supported by NSFC through Grants 91230110, 11571290, and 41375115. The authors are grateful to the anonymous referee for the valuable comments in improving the quality of the paper.

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu, China

    Changna Lu

  2. Department of Mathematics, The University of Kansas, Lawrence, KS, 66045, USA

    Weizhang Huang

  3. School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling & High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, China

    Jianxian Qiu

Authors
  1. Changna Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weizhang Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianxian Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changna Lu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, C., Huang, W. & Qiu, J. An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation. J Sci Comput 74, 122–144 (2018). https://doi.org/10.1007/s10915-017-0427-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0427-6

Keywords

Mathematics Subject Classification

Search

Navigation