Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T06:02:20.589Z Has data issue: false hasContentIssue false
  • English
  • Français

A High-Order NVD/TVD-Based Polynomial Upwind Scheme for the Modified Burgers’ Equations

Published online by Cambridge University Press:  03 June 2015

Wei Gao ,
Yang Liu ,
Bin Cao  and
Hong Li
Wei Gao*
Affiliation:
School of Mathematical Sciences, Inner Mongolia University, Huhhot, P. R. China
Yang Liu*
Affiliation:
School of Mathematical Sciences, Inner Mongolia University, Huhhot, P. R. China
Bin Cao*
Affiliation:
School of Mathematical Sciences, Inner Mongolia University, Huhhot, P. R. China
Hong Li*
Affiliation:
School of Mathematical Sciences, Inner Mongolia University, Huhhot, P. R. China
*
Corresponding author. Email: mathgao@mail.ustc.edu.cn
Email: mathliuyang@yahoo.cn
Email: bincao@gmail.com
Email: smslh@imu.edu.cn
Get access

Abstract

A bounded high order upwind scheme is presented for the modified Burgers’ equation by using the normalized-variable formulation in the finite volume framework. The characteristic line of the present scheme in the normalized-variable diagram is designed on the Hermite polynomial interpolation. In order to suppress unphysical oscillations, the present scheme respects both the TVD (total variational diminishing) constraint and CBC (convection boundedness criterion) condition. Numerical results demonstrate the present scheme possesses good robustness and high resolution for the modified Burgers’ equation.

Keywords

65M0876M12modified Burgers’ equationTVDNVDupwind scheme
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 5 , October 2012 , pp. 617 - 635
Copyright
Copyright © Global-Science Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Harris, S. E., Sonic shocks governed by the modified Burgers’ equation, Eur. J. Appl. Math., 7(2) (1996), pp. 201–222.CrossRefGoogle Scholar
[2]Sachdev, P. L. and Srinivasa Rao, Ch., N-wave solution of modified Burgers’ equation, Appl. Math. Lett., 13 (2000), pp. 1–6.CrossRefGoogle Scholar
[3]Sachdev, P. L., Srinivasa Rao, Ch. and Enflo, B.O., Large-time asymptotics for periodic solutions of the modified Burgers’ equation, Stud. Appl. Math., 114 (2005), pp. 307–323.CrossRefGoogle Scholar
[4]Inan, I. and Ugurlu, Y., Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers’ equation, Appl.Math. Comput., 217 (2009), pp. 1294–1299.Google Scholar
[5]Ramadan, M. A. and El-Danaf, T.S., Numerical treatment for the modified Burgers’ equation, Math. Comput. Simul., 70 (2005). pp. 90–98.CrossRefGoogle Scholar
[6]Ramadan, M. A., El-Danaf, T.S. and Alaal, F., A numerical solution of the Burgers’ equation using septic B-splines, Chaos Solitons Fractals, 26 (2005). pp. 795–804.CrossRefGoogle Scholar
[7]Dağ, I., Irk, D. and Saka, B., A numerical solution of the Burgers’ equation using cubic B-splines, Appl. Math. Comput., 163 (2005). pp. 199–211.Google Scholar
[8]Dağ, I., Saka, B. and Boz, A., B-spline Galerkin methods for numerical solutions of the Burgers’ equation, Appl. Math. Comput., 166 (2005). pp. 506–522.Google Scholar
[9]Saka, B. and Dağ, I., Quartic B-spline collocationmethods to the numerical solutions of the Burgers’ equation, Chaos Solitons Fractals, 32 (2007). pp. 1125–1137.CrossRefGoogle Scholar
[10]Saka, B. and Dağ, I., A numerical study of the Burgers’ equation, J. Franklin Inst., 345 (2008). pp. 328–348.CrossRefGoogle Scholar
[11]Irk, D., Sextic B-spline collocation method for the modified Burgers’ equation, Kybernetes, 38(9) (2009), pp. 1599–1620.CrossRefGoogle Scholar
[12]Temsah, R. S., Numerical solutions for convection-diffusion equation using El-Gendi method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009). pp. 760–769.CrossRefGoogle Scholar
[13]Duan, Y., Liu, R. and Jiang, Y., Lattice Boltzmann model for the modified Burgers’ equation, Appl. Math. Comput., 202 (2008). pp. 489–497.Google Scholar
[14]Bratsos, A. G., A fourth-order numerical scheme for solving the modified Burgers’ equation, Appl. Math. Comput., 60 (2010). pp. 1393–1400.CrossRefGoogle Scholar
[15]Shyy, W., A study of finite difference approximations to steady-state, a convection-dominated flow problem, J. Comp. Phys., 57 (1985), pp. 415–438.CrossRefGoogle Scholar
[16]Leonard, B. P., A stable and accurate modelling procedure based on quadratic interpolation, Comput. Meth. Appl. Mech. Engng., 19 (1979). pp. 59–98.CrossRefGoogle Scholar
[17]Agarwal, R. K., A third-order-accurate upwind scheme for Navier-Stokes solutions at high Reynolds numbers, AIAA paper 1981-112 in 19th AIAA Aerospace Sciences Meeting, St. Louis, MO, USA, (1981).Google Scholar
[18]Shah, Abdullah, Guo, Hong and Yuan, Li, A third-order upwind compact scheme on curvilinear meshes for the incompressible Navier-Stokes equations, Commun. Comput. Phys., 5 (2009), pp. 712–729.Google Scholar
[19]Lax, P. D. and Wendroff, B., Systems of conservations laws, Commun. Pur. Appl. Math., 13 (1960). pp. 217–237.CrossRefGoogle Scholar
[20]Roe, P. L., Characteristic-based schemes for the Euler equations, Ann. Rev. Fluid Mech., 18 (1986). pp. 337–365.CrossRefGoogle Scholar
[21]Harten, A., High resolution schemes for hyperbolic conservation law, J. Comp. Phys., 49 (1983). pp. 357–4393.CrossRefGoogle Scholar
[22]Sweby, P. K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21 (1984). pp. 995–1011.CrossRefGoogle Scholar
[23]Van Leer, B., Towards the ultimate conservative difference scheme monotoinity and conservation combined in a second-order scheme, J. Comp. Phys., 14 (1974). pp. 361–370.CrossRefGoogle Scholar
[24]Van Leer, B., Upwind and high-resolution methods for compressible flow: From donor cell to residual-distribution schemes, Commun. Comput. Phys., 1 (2006). pp. 192–206.Google Scholar
[25]Gaskell, P. H. and Lau, A. K. C., Curvature-compensated convective transport: SMART, a new boundedness-perserving transport algorithm, Int. J. Numer. Meth. Fl., 8 (1988), pp. 617–641.CrossRefGoogle Scholar
[26]Leonard, B. P., Simple high-accuracy resolution program for convective modeling of discontinuities, Int. J. Numer. Meth. Fl., 8 (1988). pp. 1291–1318.CrossRefGoogle Scholar
[27]Zhu, J., A low-diffusive and oscillation-free convective scheme, Comm. Appl. Mech. Eng., 7 (1991). pp. 225–232.Google Scholar
[28]Darwish, M. S.A new high-resolution scheme based on the normalized variable formulation, Numer. Heat Trans. B, 24 (1993). pp. 353–337.CrossRefGoogle Scholar
[29]Wei, J. J., Yu, B. and Tao, W. Q., A new High-order-accurate and bounded scheme for incompressible flow, Numer. Heat Trans. B, 43 (2003). pp. 19–41.CrossRefGoogle Scholar
[30]Song, B., Liu, G. R. L., Lam, K.Y. and Amano, R.S., On a higher-order discretization scheme, Int. J. Numer. Meth. Fl., 32 (2000), pp. 881–897.3.0.CO;2-6>CrossRefGoogle Scholar
[31]Alves, M. A., Oliveira, P. J. and Pinho, F. T., A convergent and universally bounded interpolation scheme for the treatment of advection, Int. J. Numer. Meth. Fl., 41 (2003). pp. 47–75.CrossRefGoogle Scholar
[32] B. YU, Tao, W. Q., Zhang, D. S. and Wang, Q. W., Discussion on numerical stability and boundedness of convective discretized scheme, Numer. Heat Trans. B, 40 (2001). pp. 343–365.Google Scholar
[33] P. L. HOU, Tao, W. Q. and Yu, M. Z., Refinemet of the convective boundedness criterion of Gaskell and Lau, Eng. Comput., 20 (2003). pp. 1023–1043.Google Scholar
[34]Lin, C-H. and Lin, C. A., Simple high-order bounded convection scheme to model discontinuities, AIAA J., 35 (1997). pp. 563–565.CrossRefGoogle Scholar
[35]Lin, H. and Chieng, C. C., A characteristic-based flux limiter of an essentially 3rd-order flux-splitting method for hyperbolic conservation laws, Int. J. Numer. Meth. Fl., 13 (1991). pp. 287–301.CrossRefGoogle Scholar
[36]Godunov, S. K., Finite difference method for numerical computaiton of discontinous solutions of the equations of the fluid dynamcis, (in Russian), Mat. Sbornik, 47 (1959), pp. 271–306.Google Scholar
[37]Hirsch, C., Numerical Compuatation of Internal and External Flows, Wiley, New York (1990).Google Scholar
[38]Zijlema, M. and Wesseling, P., Higher order flux-limiting methods for steady-state, multidimensional, convection-dominated flow, Report 1995-131, Delft University of Technology.Google Scholar
[39]Hassanien, I. A., Salama, A. A. and Hosham, H. A., Fourth-order finite difference method for solving Burgers’ equation, Appl. Math. Comput., 170 (2005). pp. 781–800.Google Scholar
[40]Kutluay, S., Bahadir, A. R. and Özdes, A., Numerical solution of one-dimensional Burgers’ equation: explicit and exact-explicit finite difference method, J. Comp. Appl. Math., 103 (1999). pp. 251–261.CrossRefGoogle Scholar
[41]Xu, M., Wang, R., Zhang, J. and Fang, Q., A novel numerical scheme for solving Burgers’ equation, Appl. Math. Comput. (2010), (accepted).Google Scholar
[42]Ferreira, V. G., Kurokawa, F. A. and Queiroz, R. A. B., Assessment of a high-order finite difference upwind scheme for the simulation of convection-diffusion problems, Int. J. Numer. Meth. Fl., 60 (2009). pp. 1–26.CrossRefGoogle Scholar
[43]Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43 (1980), pp. 357–372.CrossRefGoogle Scholar
[44]Yang, M., Wang, Z. J., A parameter-free generalized moment limiter for high-order methods on unstructured grids, Adv. Appl. Math. Mech., 4 (2009). pp. 451–480.Google Scholar
[45]Alves, M. A., Oliveira, P. J., Pinho, F. T., A convergent and universally bounded interpolation scheme for the treatment of advection, Int. J. Numer. Meth. Fl., 41 (2003). pp. 47–75.CrossRefGoogle Scholar
[46]Ng, K. C., Yusoff, M. Z. and Ng, K., Higher-order bounded differencing schemes for compressible and incompressible flows, Int. J. Numer. Meth. Fl., 53 (2007), pp. 57–80.CrossRefGoogle Scholar