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Discontinuous Legendre Wavelet Galerkin Method for One-Dimensional Advection-Diffusion Equation

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DOI: 10.4236/am.2015.69141    2,304 Downloads   2,732 Views   Citations

ABSTRACT

This paper presents discontinuous Legendre wavelet Galerkin (DLWG) approach for solving one-dimensional advection-diffusion equation (ADE). Variational formulation of this type equation and corresponding numerical fluxes are devised by utilizing the advantages of both the Legendre wavelet bases and discontinuous Galerkin (DG) method. The distinctive features of the proposed method are its simple applicability for a variety of boundary conditions and able to effectively approximate the solution of PDEs with less storage space and execution. The results of a numerical experiment are provided to verify the efficiency of the designed new technique.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Zheng, X. and Wei, Z. (2015) Discontinuous Legendre Wavelet Galerkin Method for One-Dimensional Advection-Diffusion Equation. Applied Mathematics, 6, 1581-1591. doi: 10.4236/am.2015.69141.

References

[1] Murray, J.D. (2002) Mathematical Biology I: An Introduction. Vol. 17 of Interdisciplinary Applied Mathematics, 3rd Edition, Springer, New York.
[2] Wang, X.Y., Zhu, Z.S. and Lu, Y.K. (1990) Solitary Wave Solutions of the Generalised Burgers-Huxley Equation, Journal of Physics A, 23, 271-274.
http://dx.doi.org/10.1088/0305-4470/23/3/011
[3] Isenberg, J. and Gutfinger, C. (1973) Heat Transfer to a Draining Film. International Journal of Heat and Mass Transfer, 16, 505-512.
http://dx.doi.org/10.1016/0017-9310(73)90075-6
[4] Dehghan, M. (2004) Weighted Finite Difference Techniques for One Dimensional Advection-Difffusion Equation. Applied Mathematics and Computation, 147, 307-319.
http://dx.doi.org/10.1016/S0096-3003(02)00667-7
[5] Karahan, H. (2007) Unconditional Stable Explicit Finite Difference Technique for the Advection-Diffusion Equation Using Spreadsheets. Advances in Engineering Software, 38, 80-84.
http://dx.doi.org/10.1016/j.advengsoft.2006.08.001
[6] Liang, D. and Zhao, W. (1997) A High-Order Upwind Method for the Convection-Diffusion Problem. Computer Methods in Applied Mechanics and Engineering, 147, 105-115.
http://dx.doi.org/10.1016/S0045-7825(97)00004-2
[7] Dehghan, M. (2005) On the Numerical Solution of the One Dimensional Convection-Diffusion Equation. Mathematical Problems in Engineering, 1, 61-74.
http://dx.doi.org/10.1155/MPE.2005.61
[8] Golbabai, A. and Javidi, M. (2009) A Spectral Domain Decomposition Approach for the Generalized Burger’s-Fisher Equation. Chaos, Solitons & Fractals, 39, 385-392.
http://dx.doi.org/10.1016/j.chaos.2007.04.013
[9] Mohammadi, A., Manteghian, M. and Mohammadi, A. (2011) Numerical Solution of the One-Dimensional Advection-Diffusion Equation Using Simultaneously Temporal and Spatial Weighted Parameters. Australian Journal of Basic and Applied Sciences, 5, 1536-1543.
[10] Kumar, A., Jaiswal, D.K. and Kumar, N. (2010) Analytical Solutions to One-Dimensional Advection-Diffusion Equation with Variable Coefficients in Semi-Infinite Media. Journal of Hydrology, 380, 330-337.
http://dx.doi.org/10.1016/j.jhydrol.2009.11.008
[11] Liu, H. and Yan, J. (2009) The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems. SIAM Journal on Numerical Analysis, 47, 475-694.
http://dx.doi.org/10.1137/080720255
[12] Kumar, A., Jaiswal, D.K. and Kumar, N. (2009) Analytical Solutions of One-Dimensional Advection-Diffusion Equation with Variable Coefficients in a Finite Domain. Journal of Earth System Science, 118, 539-549.
http://dx.doi.org/10.1007/s12040-009-0049-y
[13] Liu, H. and Yan, J. (2010) The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections. Communications in Computational Physics, 8, 541-564.
http://dx.doi.org/10.4208/cicp.010909.011209a
[14] Zhang, R.-P. and Zhang, L.-W. (2012) Direct Discontinuous Galerkin Method for the Generalized Burgers Fisher Equation. Chinese Physics B, 9, Article ID: 090206.
http://dx.doi.org/10.1088/1674-1056/21/9/090206
[15] Dahmen, W., Kurdila, J. and Oswald, P. (1997) Multiscale Wavelet Methods for Partial Differential Equations. Academic Press, San Diego.
[16] Razzaghi, M. and Yousefi, S. (2001) The Legendre Wavelets Operational Matrix of Integration. International Journal of Systems Science, 32, 495-502.
http://dx.doi.org/10.1080/00207720120227
[17] Yousefi, S. (2010) Legendre Multiwavelet Galerkin Method for Solving the Hyperbolic Telegraph Equation. Numerical Methods for Partial Differential Equations, 26, 539-543.
[18] Gantumur, T. and Stevenson, R.P. (2006) Computation of Differential Operators in Wavelet Coordinates. Mathematics of Computation, 75, 697-710.
http://dx.doi.org/10.1090/S0025-5718-05-01807-7
[19] Aziz, I., Siraj-ul-Islam and Sarler, B. (2013) Wavelets Collocation Method for Numerical Solution of Elliptic Problems. Applied Mathematical Modelling, 37, 676-694.
http://dx.doi.org/10.1016/j.apm.2012.02.046
[20] Liandrat, J., Perrier, V. and Tchmitchian, Ph. (1992) Numerical Resolution of Nonlinear Partial Differential Equations Using Wavelet Approach. In: Ruskai, M., Beylkin, G., Coifman, R., Daubechies, I., Mallat, S., Meyer, Y. and Raphael, L., Eds., Wavelets and Their Applications, Jones and Bartlett, Boston, 227-238.
[21] Kumar, R.B.V. and Mehra, M. (2006) A Three-Step Wavelet Galerkin Method for Parabolic and Hyperbolic Partial Differential Equations. International Journal of Computer Mathematics, 83, 143-157.
http://dx.doi.org/10.1080/00207160500112985
[22] Alpert, B., Beylkin, G., Gines, D. and Vozovoi, L. (2002) Adaptive Solution Partial Differential Equations in Multiwavelet Bases. Journal of Computational Physics, 182, 149-190.
http://dx.doi.org/10.1006/jcph.2002.7160
[23] Berlkin, G. (1992) On the Representation of Operators in Bases of Compactly Supported Wavelets. SIAM Journal on Numerical Analysis, 6, 1716-1739.
http://dx.doi.org/10.1137/0729097
[24] Zheng, X., Yang, X., Su, H. and Qiu, L. (2011) Discontinuous Legendre Wavelet Element Method for Elliptic Partial Differential Equations. Applied Mathematics and Computation, 218, 3002-3018.
http://dx.doi.org/10.1016/j.amc.2011.08.045
[25] Kima, M.-Y. and Wheelerb, M.F. (2014) A Multiscale Discontinuous Galerkin Method for Convection-Diffusion-Reaction Problems. Computers and Mathematics with Applications, 68, 2251-2261.
http://dx.doi.org/10.1016/j.camwa.2014.08.007

  
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