Science China Technological Sciences

, Volume 57, Issue 7, pp 1285–1292 | Cite as

Wavelet solutions of Burgers’ equation with high Reynolds numbers

  • XiaoJing Liu
  • YouHe Zhou
  • Lei Zhang
  • JiZeng WangEmail author
Article Special Topic: Computational Mechanics


A wavelet method is proposed to solve the Burgers’ equation. Following this method, this nonlinear partial differential equation is first transformed into a system of ordinary differential equations using the modified wavelet Galerkin method recently developed by the authors. Then, the classical fourth-order explicit Runge-Kutta method is employed to solve the resulting system of ordinary differential equations. Such a wavelet-based solution procedure has been justified by solving two test examples: results demonstrate that the proposed method has a much better accuracy and efficiency than many other existing numerical methods, and whose order of convergence can go up to 5. Most importantly, our results also indicate that the present wavelet method can readily deal with those fluid dynamics problems with high Reynolds numbers.


modified wavelet Galerkin method Runge-Kutta method Burgers’ equation high Reynolds number 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bateman H. Some recent researches on the motion of fluids. Mon Weather Rev, 1915, 43: 163–170CrossRefGoogle Scholar
  2. 2.
    Burgers J M. Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Trans Roy Neth Acad Sci Amsterdam, 1939, 17: 1–53MathSciNetGoogle Scholar
  3. 3.
    Burgers J M. A mathematical model illustrating the theory of turbulence. Adv Appl Mech, 1949, 1: 171–199CrossRefGoogle Scholar
  4. 4.
    Lai L L, Cheng R J, Li Z P, et al. The KdV-Burgers equation in a modified speed gradient continuum model. Chin Phys B, 2013, 22: 060511CrossRefGoogle Scholar
  5. 5.
    Cole J D. On a quasilinear parabolic equations occurring in aerodynamics. Q Appl Math, 1951, 9: 225–236zbMATHGoogle Scholar
  6. 6.
    Hopf E. The partial differential equation u t+uu xu xx. Commun Pure Appl Math, 1950, 3: 201–230CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Benton E, Platzman G W. A table of solutions of the one-dimensional Burgers’ equations. Q Appl Math, 1972, 30: 195–212zbMATHMathSciNetGoogle Scholar
  8. 8.
    Kutluay S, Esen A, Dag I. Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method. J Comput Appl Math, 2004, 167: 21–33CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hassanien I A, Salama A A, Hosham H A. Fourth-order finite difference method for solving Burgers’ equation. Appl Math Comput, 2005, 170: 781–800CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Saka B, Dag I. Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation. Chaos Soliton Fract, 2007, 32: 1125–1137CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kakuda K, Tosaka N. The generalized boundary element approach to Burgers’ equation. INT J Numer Meth Eng, 1990, 29: 245–261CrossRefzbMATHGoogle Scholar
  12. 12.
    Evans D J, Abdullah A R. The group explicit method for the solution of Burger’s equation. Computing, 1984, 32: 239–253CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Gardner L R T, Gardner G A, Dogan A. A Petrov-Galerkin finite element scheme for Burgers’ equation. Arab J Sci Eng, 1997, 22: 99–109zbMATHMathSciNetGoogle Scholar
  14. 14.
    Xie H, Li D. A meshless method for Burgers’ equation using MQ-RBF and high-order temporal approximation. Appl Math Model, 2013, 37: 9215–9222CrossRefMathSciNetGoogle Scholar
  15. 15.
    Jiwari R, Mittal R C, Sharma K. K. A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation. Appl Math Comput, 2013, 219: 6680–6691CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Gao Y, Le L H, Shi B C. Numerical solution of Burgers’ equation by lattice Boltzmann method. Appl Math Comput, 2013, 219: 7685–7692CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Zhang R P, Yu X J, Zhao G Z. Modified Burgers’ equation by the local discontinuous Galerkin method. Chin Phys B, 2013, 22: 030210CrossRefGoogle Scholar
  18. 18.
    Zhang W T, Dai C Q, Chen W L. Interactions among special embed-solitons for the (3+1)-dimensional Burgers equation. Chin Phys B, 2013, 22: 040509CrossRefGoogle Scholar
  19. 19.
    Hon Y C, Mao X Z. An efficient numerical scheme for Burgers’ equation. Appl Math Comput, 1998, 95: 37–50CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Ding X, Jiu Q, He C. On a nonhomogeneous Burgers’ equation. Sci China-Math, 2001, 44: 984–993CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Liu X J, Zhou Y H, Wang X M, et al. A wavelet method for solving a class of nonlinear boundary value problems. Commun Nonlinear Sci Numer Simul, 2013, 18: 1939–1948CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Liu X J, Wang J Z, Zhou Y H. Wavelet solution of a class of two-dimensional nonlinear boundary value problems. CMES: Comp Model Eng Sci, 2013, 92: 493–505MathSciNetGoogle Scholar
  23. 23.
    Odejide S A, Aregbesola Y A S. A note on two dimensional bratu problem. Kragujevac J Math, 2006, 29: 49–56zbMATHMathSciNetGoogle Scholar
  24. 24.
    Jalilian R. Non-polynomial spline method for solving Bratu’s problem. Comput Phys Commun, 2010, 181: 1868–1872CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Caglar H, Caglar N, Özer M, et al. B-spline method for solving Bratu’s problem. Int J Comput Math, 2010, 87: 1885–1891CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Abbasbandy S, Hashemi M S, Liu C S. The Lie-group shooting method for solving the Bratu equation. Commun Nonlinear Sci Numer Simul, 2011, 16: 4238–4249CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Hassan I H A H, Ertürk V S. Applying differential transformation method to the one-dimensional planar Bratu problem. Int J Contemp Math Sci, 2007, 2: 1493–1504zbMATHMathSciNetGoogle Scholar
  28. 28.
    Khuri S A. A new approach to Bratu’s problem. Appl Math Comput, 2004, 147: 131–136CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Deeba E, Khuri S A, Xie S. An algorithm for solving boundary value problems. J Comput Phys, 2000, 159: 125–138CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Wang J Z. Generalized theory and arithmetic of orthogonal wavelets and applications to researches of mechanics including piezoelectric smart structures. Dissertation of the Doctoral Degree. Lanzhou: Lanzhou University, 2001Google Scholar
  31. 31.
    Wood W L. An exact solution for Burger’s equation. Comm Numer Meth Eng, 2006, 22: 797–798CrossRefzbMATHGoogle Scholar
  32. 32.
    Mittal R C, Jain R K. Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method. Appl Math Comput, 2012, 218: 7839–7855CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Chen R, Wu Z. Applying multiquadric quasi-interpolation to solve Burgers’ equation. Appl Math Comput, 2006, 172: 472–484CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Zhu C G, Wang R H. Numerical solution of Burgers’ equation by cubic B-spline quasi-interpolation. Appl Math Comput, 2009, 208: 260–272CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Kutluay S, Esen A, Dag I. Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method. J Comput Appl Math, 2004, 167: 21–33CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • XiaoJing Liu
    • 1
  • YouHe Zhou
    • 1
  • Lei Zhang
    • 1
  • JiZeng Wang
    • 1
    Email author
  1. 1.Key Laboratory of Mechanics on Disaster and Environment in Western China, the Ministry of Education, and School of Civil Engineering and MechanicsLanzhou UniversityLanzhouChina

Personalised recommendations