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 JAMP  Vol.3 No.5 , May 2015
High Order Compact Difference Scheme and Multigrid Method for 2D Elliptic Problems with Variable Coefficients and Interior/Boundary Layers on Nonuniform Grids
Abstract: In this paper, a high order compact difference scheme and a multigrid method are proposed for solving two-dimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a nonuniform mesh) to the computational domain (with a uniform mesh) by using a coordinate transformation. Then, a fourth order compact difference scheme is proposed to solve the transformed elliptic equation on uniform girds. After that, a multigrid method is employed to solve the linear algebraic system arising from the difference equation. At last, the numerical experiments on some elliptic problems with interior/boundary layers are conducted to show high accuracy and high efficiency of the present method.
Cite this paper: Lan, B. , Ge, Y. , Wang, Y. and Zhan, Y. (2015) High Order Compact Difference Scheme and Multigrid Method for 2D Elliptic Problems with Variable Coefficients and Interior/Boundary Layers on Nonuniform Grids. Journal of Applied Mathematics and Physics, 3, 509-523. doi: 10.4236/jamp.2015.35063.
References

[1]   Lynch, R.E. and Rice, J.R. (1978) High Accuracy Finite Difference Approximation to Solutions of Elliptic Partial Differential Equations. Proceedings of the National Academy of Sciences of the United States of America, 75, 2541-2544.
http://dx.doi.org/10.1073/pnas.75.6.2541

[2]   Boisvert, R.F. (1981) Families of High Order Accurate Discretizations of Some Elliptic Problems. SIAM Journal on Scientific and Statistical Computing, 2, 268-284.
http://dx.doi.org/10.1137/0902022

[3]   Gupta, M.M., Manohar, R.P. and Stephenson, J.W. (1984) A Single Cell High Order Scheme for the Convection Diffusion Equation with Variable Coefficients. International Journal for Numerical Methods in Fluids, 4, 641-651.
http://dx.doi.org/10.1002/fld.1650040704

[4]   Gupta, M.M., Manohar, R.P. and Stephenson, J.W. (1985) High-Order Difference Scheme for Two Dimensional Elliptic Equations. Numerical Methods for Partial Differential Equations, 1, 71-80.
http://dx.doi.org/10.1002/num.1690010108

[5]   Ananthakrishnaiah, U., Manohar, R. and Stephenson, J.W. (1987) High-Order Methods for Elliptic Equations with Variable Coefficients. Numerical Methods for Partial Differential Equations, 3, 219-227.
http://dx.doi.org/10.1002/num.1690030306

[6]   Ananthakrishnaiah, U., Manohar, R. and Stephenson, J.W. (1987) Fourth-Order Finite Difference Methods for Three-Dimensional General Linear Elliptic Problems with Variable Coefficients. Numerical Methods for Partial Differential Equations, 3, 229-240.
http://dx.doi.org/10.1002/num.1690030307

[7]   Ge, L.X. and Zhang, J. (2002) Symbolic Computation of High Order Compact Difference Schemes for Three Dimensional Linear Elliptic Partial Differential Equations with Variable Coefficients. Journal of Computational and Applied Mathematics, 143, 9-27.
http://dx.doi.org/10.1016/S0377-0427(01)00504-0

[8]   Choo, J.Y. (1994) Stable High Order Methods for Elliptic Equations with Large First Order Terms. Computers and Mathematics with Applications, 27, 65-80.
http://dx.doi.org/10.1016/0898-1221(94)90006-X

[9]   Karaa, S. (2007) High-Order Difference Scheme for 2D Elliptic and Parabolic Problems with Mixed Derivatives. Numerical Methods for Partial Differential Equations, 23, 366-378.
http://dx.doi.org/10.1002/num.20181

[10]   Briane, M. and Diaz, J.C. (2008) Uniform Convergence of Sequences of Solutions of Two-Dimensional Linear Elliptic Equations with Unbounded Coefficients. Journal of Differetial Equations, 245, 2038-2054.
http://dx.doi.org/10.1016/j.jde.2008.07.027

[11]   Choo, J.Y. and Schultz, D.H. (1994) A High Order Difference Method of Steady State Navier-Stokes Equation. Computers & Mathematics with Applications, 27, 105-119.
http://dx.doi.org/10.1016/0898-1221(94)90101-5

[12]   Spotz, W.F. (1995) High-Order Finite Difference Schemes for Computational Mechanics. Ph.D. Thesis, University of Texas at Austin.

[13]   Ge, L.X. and Zhang, J. (2001) High Accuracy Iterative Solution of Convection Diffusion Equation with Boundary Layers on Nonuniform Grids. Journal of Computational Physics, 171, 560-578.
http://dx.doi.org/10.1006/jcph.2001.6794

[14]   Zhang, J., Ge, L.X. and Gupta, M.M. (2001) Fourth Order Compact Difference Scheme for 3D Convection Diffusion Equation with Boundary Layers on Nonuniform Grids. Neural, Parallel and Scientific Computations, 8, 373-392.

[15]   Liu, C. and Liu, Z. (1995) Multigrid Mapping and Box Relaxation for Simulation of the Whole Process of Flow Transition in 3D Boundary Layers. Journal of Computational Physics, 119, 325-341.
http://dx.doi.org/10.1006/jcph.1995.1138

[16]   Brandt, A. (1977) Multi-Level Adaptive Solution to Boundary Value Problems. Mathematics of Computation, 31, 333-390.
http://dx.doi.org/10.1090/S0025-5718-1977-0431719-X

[17]   Hackbusch, W. and Trottenberg, U. (1982) Multigrid Methods. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/BFb0069927

[18]   Wesseling, P. (1992) An Introduction to Multigrid Methods. Wiley, Chichester.

[19]   Gupta, M.M., Kouatchou, J. and Zhang, J. (1997) Comparison of Second- and Fourth-Order Discretizations for Multigrid Poisson Solvers. Journal of Computational Physics, 132, 226-232.
http://dx.doi.org/10.1006/jcph.1996.5466

[20]   Zhang, J. (1998) Fast and High Accuracy Multigrid Solution of the Three Dimensional Poisson Equation. Journal of Computational Physics, 143, 449-461.
http://dx.doi.org/10.1006/jcph.1998.5982

[21]   Ge, Y., Cao, F. and Zhang, J. (2013) A Transformation-Free HOC Scheme and Multigrid Method for Solving the 3D Poisson Equation on Nonuniform Grids. Journal of Computational Physics, 234, 199-216.
http://dx.doi.org/10.1016/j.jcp.2012.09.034

[22]   Gupta, M.M., Kouatchou, J. and Zhang, J. (1997) A Compact Multigrid Solver for Convection-Diffusion Equations. Journal of Computational Physics, 132, 123-129.
http://dx.doi.org/10.1006/jcph.1996.5627

[23]   Zhang, J. (1997) Accelerated Multigrid High Accuracy Solution of the Convection-Diffusion Equation with High Reynolds Number. Numerical Methods for Partial Differential Equations, 13, 77-92.
http://dx.doi.org/10.1002/(SICI)1098-2426(199701)13:1<77::AID-NUM6>3.0.CO;2-J

[24]   Gupta, M.M. and Zhang, J. (2000) High Accuracy Multigrid Solution of the 3D Convection-Diffusion Equation. Applied Mathematics and Computation, 113, 249-274.
http://dx.doi.org/10.1016/S0096-3003(99)00085-5

[25]   Ge, Y. and Cao, F. (2011) Multigrid Method Based on the Transformation-Free HOC Scheme on Nonuniform Grids for 2D Convection Diffusion Problems. Journal of Computational Physics, 230, 4051-4070.
http://dx.doi.org/10.1016/j.jcp.2011.02.027

[26]   Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Rordan, E. and Shishkin, G.I. (2000) Robust Computational Techniques for Boundary Layers. Chapman & Hall/CRC, Boca Raton.

 
 
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