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A High-Accuracy Finite Difference Scheme for Solving Reaction-Convection-Diffusion Problems with a Small Diffusivity

  • Po-Wen Hsieh (a1), Suh-Yuh Yang (a2) and Cheng-Shu You (a2)

This paper is devoted to a new high-accuracy finite difference scheme for solving reaction-convection-diffusion problems with a small diffusivity £. With a novel treatment for the reaction term, we first derive a difference scheme of accuracy O(ɛ2h+ɛh2+h3) for the 1-D case. Using the alternating direction technique, we then extend the scheme to the 2-D case on a nine-point stencil. We apply the high-accuracy finite difference scheme to solve the 2-D steady incompressible Navier-Stokes equations in the stream function-vorticity formulation. Numerical examples are given to illustrate the effectiveness of the proposed difference scheme. Comparisons made with some high-order compact difference schemes show that the newly proposed scheme can achieve good accuracy with a better stability.

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[1] H. Ding and Y. Zhang , A new difference scheme with high accuracy and absolute stability for solving convection-diffusion equations, J. Comput. Appl. Math., 230 (2009), pp. 600606.

[3] U. Ghia , K. N. Ghia and C. T. Shin , High Re-solution for incompressible Navier-Stokes equation and a multigrid method, J. Comput. Phys., 48 (1982), pp. 387411.

[4] M. M. Gupta , High accuracy solutions of incompressible Navier-Stokes equations, J. Comput. Phys., 93 (1991), pp. 343359.

[5] M. M. Gupta and J. C. Kalita , A new paradigm for solving Navier-Stokes equations: streamfunction-velocity formulation, J. Comput. Phys., 207 (2005), pp. 5268.

[6] P.-W. Hsieh and S.-Y. Yang , Two new upwind difference schemes for a coupled system of convection-diffusion equations arising from the steady MHD duct flow problems, J. Comput. Phys., 229 (2010), pp. 92169234.

[7] P.-W. Hsieh and S.-Y. Yang , A novel least-squaresfinite element method enriched with residualfree bubbles for solving convection-dominated problems, SIAM J. Sci. Comput., 32 (2010), pp. 20472073.

[8] S. Karaa and J. Zhang , High order ADI method for solving unsteady convection-diffusion problems, J. Comput. Phys., 198 (2004), pp. 19.

[9] S. Karaa , A hybrid Pade ADI scheme of higher-order for convection-diffusion problems, Int. J. Numer. Meth. Fluids, 64 (2010), pp. 532548.

[10] R. J. LeVeque , Finite Difference Methods for Ordinary and Partial Differential Equations, Society for Industrial and Applied Mathematics, Philadelphia, USA, 2007.

[11] M. Li , T. Tang and B. Fornberg , A compact fourth-order finite difference scheme for the steady incompressibe Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 20 (1995), pp. 11371151.

[13] K. Mahesh , A family of high order finite difference schemes with good spectral resolution, J. Comput. Phys., 145 (1998), pp. 332358.

[15] E. O’Riordan and M. Stynes , Numerical analysis of a strongly coupled system of two singularly perturbed convection-diffusion problems, Adv. Comput. Math., 30 (2009), pp. 101121.

[16] A. C. Radhakrishna Pillai , Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type, Int. J. Numer. Meth. Fluids, 37 (2001), pp. 87106.

[17] H.-G. Roos , M. Stynes and L. Tobiska , Numerical Methods for Singularly Perturbed Differential Equations, Springer, New York, 1996.

[18] Y. V. S. S. Sanyasiraju and N. Mishra , Spectral resolutioned exponential compact higher order scheme (SRECHOS) for convection-diffusion equations, Comput. Methods Appl. Mech. Eng., 197 (2008), pp. 47374744.

[21] W. F. Spotz , Accuracy and performance of numerical wall boundary conditions for steady, 2D, incompressible streamfunction vorticity, Int. J. Numer. Meth. Fluids, 28 (1998), pp. 737757.

[22] W. F. Spotz and G. F. Carey , High-order compact scheme for the stream-function vorticity equations, Int. J. Numer. Meth. Eng., 38 (1995), pp. 34973512.

[23] M. Stynes , Steady-state convection-diffusion problems, Acta Numer., (2005), pp. 445508.

[24] Z. F. Tian and S. Q. Dai , High-order compact exponential finite difference methods for convection-diffusion type problems, J. Comput. Phys., 220 (2007), pp. 952974.

[25] 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., 198 (2007), pp. 268286.

[26] I. Yavneh , Analysis of a fourth-order compact scheme for convection-diffusion, J. Comput. Phys., 133 (1997), pp. 361364.

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Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
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