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Solution of the Advection-diffusion Equation using the Differential Quadrature Method

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KSCE Journal of Civil Engineering Aims and scope

Abstract

Differential Quadrature Method (DQM) to integrate the one-dimensional Advection-diffusion Equation (ADE) is presented. This method was applied to two examples and the results were compared with the performance of the Explicit Finite Difference Method (EFDM) and Implicit Finite Differences Method (IFDM). Based on the comparison with the exact solution, and both the explicit and implicit finite difference solutions, it was concluded that the DQM provides similar results but less grid points; besides the results are converged quickly. A numerical comparison for a case in which the exact solution is known, DQM gives closer results to the exact values than EFDM and IFDM.

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References

  • Al-Saif, A. S. J. and Zheng-You, Z. (2003). “Application of mixed differential quadrature method for solving the coupled two-dimensional incompressible Navier-stokes equation and heat equation.” Journal of Shanghai University (English Edition), Vol. 7, No. 4, pp. 343–351.

    Article  MATH  MathSciNet  Google Scholar 

  • Baeumer, B., Benson, D. A., Meerschaert, M. M., and Wheatcraft, S. W. (2001). “Subordinated advection-dispersion equation for contaminant transport.” Water Resources Research, Vol. 37, No. 6, pp. 1543–1550.

    Article  Google Scholar 

  • Bajracharya, K. and Barry, D. A. (1997). “Accuracy criteria for linearized diffusion wave flood routing.” Journal of Hydrology, Vol. 195, pp. 200–217.

    Article  Google Scholar 

  • Bellman, R. and Casti, J. (1971). “Differential quadrature and long-term integration.” Journal of Mathematical Analysis and Applications. Vol. 34,Issue 2, pp. 235–238.

    Article  MATH  MathSciNet  Google Scholar 

  • Bert, C. W. and Malik, M. (1996). “Differential quadrature method in computational mechanics: A review.” Applied Mechanics Review, Vol. 49, No. 1, pp. 1–28.

    Article  Google Scholar 

  • Civalek, Ö. (2003). Linear and non-linear dynamic analysis of multi-degree-of-freedom systems by harmonic differential quadrature method (in Turkish), PhD Thesis, Izmir, Dokuz Eylul University, Graduate School of Natural and Applied Sciences.

    Google Scholar 

  • Civalek, Ö. (2004). “Application of Dierential Quadrature (DQ) and Harmonic Dierential Quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns.” Engineering Structures, Elsevier, Vol. 26,Issue 2, pp. 171–186.

    Article  Google Scholar 

  • Dehghan, M. (2004). “Weighted nite difference techniques for the one-dimensional advection-diffusion equation.” Applied Mathematics and Computation, Vol. 147,Issue 2, pp. 307–319.

    Article  MATH  MathSciNet  Google Scholar 

  • Enrique, P., Emilio, D., Rosell’o, M. D., and Romero, J. V. (2008). “A stable numerical method for solving variable coefficient advection-diffusion models.” Computers and Mathematics with Applications, Vol. 56,Issue 3, pp. 754–768.

    MATH  MathSciNet  Google Scholar 

  • Gane, C. R. and Stephenson, P. L. (1979). “An explicit numerical method for solving transient combined heat conduction and convection problems.” Int. J. Numer. Meth. Eng., Vol. 14, No. 8, pp. 1141–1163.

    Article  MATH  Google Scholar 

  • Geiser, J. (2007). “Analytical solutions for convection-diffusion-dispersion-reaction-equations with different retardation-factors and applications in 2d and 3d.” Hikari Ltd., International Mathematical Forum, Vol. 2, No. 44, pp. 2179–2205.

    MATH  MathSciNet  Google Scholar 

  • Hongxing, R. (2008). “A conservative characteristic finite volume element method for solution of the advection-diffusion equation.” Comput. Methods Appl. Mech. Engrg., Vol. 197, pp. 3862–3869.

    Article  MathSciNet  Google Scholar 

  • Park, Y. S. and Baik, J. J. (2008). “Analytical solution of the advection-diffusion equation for a ground-level finite area source.” Atmospheric Environment, doi: 10.1016/j.atmosenv.2008.09.019

  • Revelli, R. and Ridolfi, L. (2008). “Generalized collocation method for two-dimensional reaction-diffusion problems with homogeneous Neumann boundary conditions.” Computers and Mathematics with Applications, Elsevier, Vol. 56,Issue 9, pp. 2360–2370.

    Article  MATH  MathSciNet  Google Scholar 

  • Roop, J. P. (2008). “Numerical approximation of a one-dimensional space fractional advection-dispersion equation with boundary layer.” Computers and Mathematics with Applications, Vol. 56,Issue 7, pp. 1808–1819.

    Article  MATH  MathSciNet  Google Scholar 

  • Schlegel, M., Knotha, O., Arnoldb, M., and Wolke, R. (2008). “Multirate Runge-Kutta schemes for advection equations.” Journal of Computational and Applied Mathematics, doi:10.1016/j.cam. 2008.08.009

  • Shu, C. (2000). Dierential quadrature and its application in engineering, Springer-Verlag London Limited.

  • Shu, C. and Chew, Y. T. (1997). “Fourier expansion-based differential quadrature and its application to Helmholtz eigenvalue problems.” Communications in Numerical Methods in Engineering, Vol. 13, pp. 643–653.

    Article  MATH  MathSciNet  Google Scholar 

  • Shu, C. and Richards, B. E. (1992). “Application of generalized differential quadrature to solve two-dimensional incompressible navierstokes equations.” International Journal for Numerical Methods in Fluids, Vol. 15,Issue 7, pp. 791–798.

    Article  MATH  Google Scholar 

  • Shu, C., Wang, L., Chew, Y. T., and Zhao, N. (2004). “Numerical study of eccentric Couette-Taylor flows and effect of eccentricity on flow patterns.” Theoret. Comput. Fluid Dynamics, Vol. 18, No. 1, pp. 43–59.

    Article  MATH  Google Scholar 

  • Shu, C. and Xue, H. (1997). “Explicit computations of weighting coefficients in the harmonic differential quadrature.” Journal of Sound and Vibration, Vol. 204, No. 3, pp. 549–555.

    Article  Google Scholar 

  • Shu, C., Yao, Q., Yeo, K. S., and Zhu, Y. D. (2002). “Numerical analysis of flow and thermal fields in arbitrary eccentric annulus by differential quadrature method.” Heat and Mass Transfer, Springer-Verlag, Vol. 38,No. 7–8, pp. 597–608.

    Article  Google Scholar 

  • Temsah, R. S. (2009). “Numerical solutions for convection-diffusion equation using El-Gendi method.” Communications in Nonlinear Science and Numerical Simulation, Vol. 14,Issue 3, pp. 760–769.

    Article  MathSciNet  Google Scholar 

  • Tian, Z. F. and Dai, S. Q. (2007). “High-order compact exponential finite difference methods for convection-diffusion type problems.” Journal of Computational Physics, Vol. 220,Issue 2, pp. 952–974.

    Article  MATH  MathSciNet  Google Scholar 

  • Yeh, H. D. and Yeh, G. T. (2007). “Analysis of point-source and boundary-source solutions of one-dimensional groundwater transport equation.” ASCE Journal of Environmental Engineering, ASCE, Vol. 133, No. 11, pp. 1032–1041.

    Article  Google Scholar 

  • Yen, B. C. and Tsai, C. W. (2001). “On noninertia wave versus diffusion wave in flood routing.” Journal of Hydrology, Vol. 244, pp. 97–104.

    Article  Google Scholar 

  • Younes, A. and Ackerer, P. (2005). “Solving the advection-diusion equation with the Eulerian-Lagrangian localized adjoint method on unstructured meshes and non uniform time stepping.” Journal of Computational Physics, Elsevier,Vol. 208,Issue 1, pp. 384–402.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Civil Engineering Department, Dokuz Eylül University, Buca, Izmir, Turkey

    Birol Kaya

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  1. Birol Kaya
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Kaya, B. Solution of the Advection-diffusion Equation using the Differential Quadrature Method. KSCE J Civ Eng 14, 69–75 (2010). https://doi.org/10.1007/s12205-010-0069-9

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  • DOI: https://doi.org/10.1007/s12205-010-0069-9

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