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Numerical Heat Transfer, Part B: Fundamentals
An International Journal of Computation and Methodology
Volume 70, 2016 - Issue 1
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Original Articles

New unconditionally stable scheme for solving the convection–diffusion equation based on the Associated Hermite orthogonal functions

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Pages 80-90 | Received 05 Dec 2015, Accepted 20 Feb 2016, Published online: 23 Jun 2016

References

  • B. J. Noye and H. H. Tan, Finite Difference Methods for Solving the Two-dimensional Advection–Diffusion Equation, Int. J. Numer. Meth. Fluids, vol. 9, no. 1, pp. 75–98, 1988.
  • L. Portero and J. C. Jorge, A Generalization of Peaceman–Rachford Fractional Step Method, J. Comput. Appl. Math., vol. 189, pp. 676–688, 2006.
  • S. M. Dash and T. S. Lee, Natural Convection in a Square Enclosure with a Square Heat Source at Different Horizontal and Diagonal Eccentricities, Numer. Heat Transfer A, vol. 68, no. 6, pp. 686–710, 2015.
  • H. S. Yoon, Y. G. Park, and J. H. Jung, Natural Convection in a Square Enclosure with Differentially Heated Two Horizontal Cylinders, Numer. Heat Transfer A, vol. 65, no. 4, pp. 302–326, 2014.
  • F. Wu, G. Wang, and W. Zhou, A Thermal Nonequilibrium Approach to Natural Convection in a Square Enclosure Due to the Partially Cooled Sidewalls of the Enclosure, Numer. Heat Transfer A, vol. 67, no. 7, pp. 771–790, 2015.
  • D. W. Peaceman and H. H. Rachford, The Numerical Solution of Parabolic and Elliptic Differential Equations, J. Soc. Ind. Appl. Math., vol. 3, no.1, pp. 28–41, 1955.
  • Y. N. Zhang and Z. Z. Sun, Alternating Direction Implicit Schemes for the Two-dimensional Fractional Sub-diffusion Equation, J. Comput. Phys., vol. 230, no. 24, pp. 8713–8728, 2011.
  • C. M. Chen, F. Liu, I. Turner, and V. Anh, Numerical Schemes and Multivariate Extrapolation of a Two-dimensional Anomalous Sub-diffusion Equation, Numer. Algorithms, vol. 54, no. 1, pp. 1–21, 2010.
  • M. Cui, Compact Alternating Direction Implicit Method for Two-dimensional Time Fractional Diffusion Equation, J. Comput. Phys., vol. 231, no. 6, pp. 2621–2633, 2012.
  • M. R. Cui, Convergence Analysis of High-order Compact Alternating Direction Implicit Schemes for the Two-dimensional Time Fractional Diffusion Equation, Numer. Algorithms, vol. 62, no. 3, pp. 383–409, 2013.
  • S. Karaa, A Hybrid Padé ADI Scheme of Higher-order for Convection–Diffusion Problems, Int. J. Numer. Meth. Fluids, vol. 64, no. 5, pp. 532–548, 2010.
  • S. Y. Zhai, X. L. Feng, and Y. N. He, A New High-order Compact ADI Method for 3-D Unsteady Convection–Diffusion Problems with Discontinuous Coefficients, Numer. Heat Transfer B, vol. 65, no. 4, pp. 376–391, 2014.
  • W. Z. Dai, A New Accurate Finite Difference Scheme for Neumann (Insulated) Boundary Condition of Heat Conduction, Int. J. Therm. Sci., vol. 49, no. 3, pp. 571–579, 2010.
  • Y. Damrongsak and M. Nikolay, Deferred Correction Technique to Construct High-order Schemes for the Heat Equation with Dirichlet and Neumann Boundary Conditions, Eng. Lett., vol. 21, no. 2, pp. 61–67, 2013.
  • Z. F. Tian and Y. B. Ge, A Fourth-order Compact ADI Method for Solving Two-dimensional Unsteady Convection–Diffusion Problems, J. Comput. Appl. Math., vol. 198, no. 1, pp. 268–286, 2007.
  • J. C. Kalita, D. C. Dalal, and A. K. Dass, A Class of Higher Order Compact Schemes for the Unsteady Convection–Diffusion Equation with Variable Convection Coefficients, Int. J. Numer. Meth. Fluids, vol. 38, no.12, pp. 1111–1131, 2002.
  • S. Karaa and J. Zhang, High Order ADI Method for Solving Unsteady Convection–Diffusion Problems, J. Comput. Phys., vol. 198, no. 1, pp. 1–9, 2004.
  • N. Li, B. Meng, X. Feng, and D. Gui, The Spectral Collocation Method for the Stochastic Allen–Cahn Equation via Generalized Polynomial Chaos, Numer. Heat Transfer B, vol. 68, no. 1, pp. 11–29, 2015.
  • Y. Sun, J. Ma, B. Li, and Z. Guo, Predication of Nonlinear Heat Transfer in a Convective–Radiative Fin with Temperature-dependent Properties by the Collocation Spectral Method, Numer. Heat Transfer B, vol. 69, no. 1, pp. 68–83, 2016.
  • S. V. Diwakar, S. K. Das, and T. Sundararajan, Accurate Solutions of Rayleigh–Bénard Convection in Confined Two-layer Systems using the Spectral Domain Decomposition Method, Numer. Heat Transfer A, vol. 66, no. 11, pp. 1218–1242, 2014.
  • D. Funaro and O. Kavian. Approximation of Some Diffusion Evolution Equations in Unbounded Domains by Hermite Functions, Math. Comput., vol. 57, no. 196, pp. 597–619, 1991.
  • B. Y. Guo, Error Estimation of Hermite Spectral Method for Nonlinear Partial Differential Equations, Math. Comput., vol. 198, no. 227, pp. 1067–1078, 1999.
  • B. Y. Guo and C. L. Xu, Hermite Pseudospectral Method for Nonlinear Partial Differential Equations. ESAIM Math. Model. Numer. Anal., vol. 34, no. 4, pp. 859–872, 2000.
  • H. Ma, W. Sun, and T. Tang, Hermite Spectral Methods with a Time-dependent Scaling for Parabolic Equations in Unbounded Domains, SIAM J. Numer. Anal., vol. 43, no. 1, pp. 58–75, 2005.
  • H. Ma and T. Zhao, A Stabilized Hermite Spectral Method for Second-order Differential Equations in Unbounded Domains, Numer. Meth. D. E., vol. 23, no. 5, pp. 968–983, 2007.
  • X. Luo, S. T. Yau, and S. S. T. Yau, Time-dependent Hermite–Galerkin Spectral Method and its Applications, Appl. Math. Comput., vol. 264, pp. 378–391, 2015.
  • J. Shen and T. Tao, Spectral and High-order Methods with Applications, Science Press, Beijing, 2006.

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