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High-order compact difference scheme and multigrid method for solving the 2D elliptic problems. (English) Zbl 1427.65333

Summary: A high-order compact difference scheme for solving the two-dimensional (2D) elliptic problems is proposed by including compact approximations to the leading truncation error terms of the central difference scheme. A multigrid method is employed to overcome the difficulties caused by conventional iterative methods when they are used to solve the linear algebraic system arising from the high-order compact scheme. Numerical experiments are conducted to test the accuracy and efficiency of the present method. The computed results indicate that the present scheme achieves the fourth-order accuracy and the effect of the multigrid method for accelerating the convergence speed is significant.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations

Software:

Wesseling
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Full Text: DOI

References:

[1] Carey, G. F.; Spotz, W. F., Higher-order compact mixed methods, Communications in Numerical Methods in Engineering, 13, 7, 553-564 (1997) · Zbl 0885.65109 · doi:10.1002/(SICI)1099-0887(199707)13:7<553::AID-CNM80>3.0.CO;2-O
[2] Gupta, M. M., A survey of some second-order difference schemes for the steady-state convection-diffusion equation, International Journal for Numerical Methods in Fluids, 3, 4, 319-331 (1983) · Zbl 0521.76090 · doi:10.1002/fld.1650030403
[3] Gupta, M. M.; Manohar, R. P.; Stephenson, J. W., A single cell high order scheme for the convection-diffusion equation with variable coefficients, International Journal for Numerical Methods in Fluids, 4, 7, 641-651 (1984) · Zbl 0545.76096 · doi:10.1002/fld.1650040704
[4] Gupta, M. M.; Manohar, R. P.; Stephenson, J. W., High-order difference schemes for two-dimensional elliptic equations, Numerical Methods for Partial Differential Equations, 1, 1, 71-80 (1985) · Zbl 0634.65078 · doi:10.1002/num.1690010108
[5] Karaa, S., High-order difference schemes for 2D elliptic and parabolic problems with mixed derivatives, Numerical Methods for Partial Differential Equations, 23, 2, 366-378 (2007) · Zbl 1112.65082 · doi:10.1002/num.20181
[6] Spotz, W. F.; Carey, G. F., A high-order compact formulation for the 3d poisson equation, Numerical Methods for Partial Differential Equations, 12, 2, 235-243 (1996) · Zbl 0866.65066 · doi:10.1002/(SICI)1098-2426(199603)12:2<235::AID-NUM6>3.3.CO;2-2
[7] Sutmann, G.; Steffen, B., High-order compact solvers for the three-dimensional Poisson equation, Journal of Computational and Applied Mathematics, 187, 2, 142-170 (2006) · Zbl 1081.65099 · doi:10.1016/j.cam.2005.03.041
[8] Gupta, M. M.; Kouatchou, J., Symbolic Derivation of Finite Difference Approximations for the Three-Dimensional Poisson Equation, Numerical Methods for Partial Differential Equations, 14, 5, 593-606 (1998) · Zbl 0926.65103 · doi:10.1002/(SICI)1098-2426(199809)14:5<593::AID-NUM4>3.0.CO;2-D
[9] Zhai, S.; Feng, X.; He, Y., A family of fourth-order and sixth-order compact difference schemes for the three-dimensional Poisson equation, Journal of Scientific Computing, 54, 1, 97-120 (2013) · Zbl 1259.65160 · doi:10.1007/s10915-012-9607-6
[10] Zhai, S.; Feng, X.; He, Y., A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation, Applied Mathematics and Computation, 230, 9-26 (2014) · Zbl 1410.65420 · doi:10.1016/j.amc.2013.12.096
[11] Zhai, S.; Feng, X.; Liu, D., A novel method to deduce a high-order compact difference scheme for the three-dimensional semilinear convection-diffusion equation with variable coefficients, Numerical Heat Transfer, Part B: Fundamentals, 63, 5, 425-455 (2013) · doi:10.1080/10407790.2013.778628
[12] Tian, Z. F.; Dai, S. Q., High-order compact exponential finite difference methods for convection-diffusion type problems, Journal of Computational Physics, 220, 2, 952-974 (2007) · Zbl 1109.65089 · doi:10.1016/j.jcp.2006.06.001
[13] Pillai, A. C., Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type, International Journal for Numerical Methods in Fluids, 37, 1, 87-106 (2001) · Zbl 1046.76031 · doi:10.1002/fld.167
[14] Zhang, J.; Sun, H.; Zhao, J. J., High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems, Computer Methods Applied Mechanics and Engineering, 191, 41-42, 4661-4674 (2002) · Zbl 1068.76066 · doi:10.1016/S0045-7825(02)00398-5
[15] Ge, L.; Zhang, J., High Accuracy Iterative Solution of Convection Diffusion Equation with Boundary Layers on Nonuniform Grids, Journal of Computational Physics, 171, 2, 560-578 (2001) · Zbl 0990.65117 · doi:10.1006/jcph.2001.6794
[16] Kalita, J. C.; Dass, A. K.; Dalal, D. C., A transformation-free {HOC} scheme for steady convection-diffusion on non-uniform grids, International Journal for Numerical Methods in Fluids, 44, 1, 33-53 (2004) · Zbl 1062.76035 · doi:10.1002/fld.621
[17] Zhang, J.; Ge, L.; Gupta, M. M., Fourth order compact difference scheme for 3D convection diffusion equation with boundary layers on non-uniform grids, Neural. Parallel & Scientific Computations, 8, 373-392 (2008) · Zbl 0983.65114
[18] Sun, H.; Zhang, J., A high-order finite difference discretization strategy based on extrapolation for convection diffusion equations, Numerical Methods for Partial Differential Equations, 20, 1, 18-32 (2004) · Zbl 1038.65108 · doi:10.1002/num.10075
[19] Ma, Y.; Ge, Y., A high order finite difference method with Richardson extrapolation for 3D convection diffusion equation, Applied Mathematics and Computation, 215, 9, 3408-3417 (2010) · Zbl 1181.65131 · doi:10.1016/j.amc.2009.10.035
[20] Karaa, S., High-order approximation of 2D convection-diffusion equation on hexagonal grids, Numerical Methods for Partial Differential Equations, 22, 5, 1238-1246 (2006) · Zbl 1098.65102 · doi:10.1002/num.20149
[21] Zhang, J.; Kouatchou, J.; Ge, L., A family of fourth-order difference schemes on rotated grid for two-dimensional convection-diffusion equation, Mathematics and Computers in Simulation, 59, 5, 413-429 (2002) · Zbl 1001.65113 · doi:10.1016/S0378-4754(01)00418-9
[22] Mohanty, R. K.; Singh, S., A new fourth order discretization for singularly perturbed two dimensional non-linear elliptic boundary value problems, Applied Mathematics and Computation, 175, 2, 1400-1414 (2006) · Zbl 1093.65103 · doi:10.1016/j.amc.2005.08.023
[23] Dai, R.; Wang, Y.; Zhang, J., Fast and high accuracy multiscale multigrid method with multiple coarse grid updating strategy for the 3D convection-diffusion equation, Computers & Mathematics with Applications. An International Journal, 66, 4, 542-559 (2013) · Zbl 1360.65259 · doi:10.1016/j.camwa.2013.06.008
[24] Sleijpen, G. L. G.; Van der Vorst, H. A.; Hafez, M.; Oshima, K., Hybrid bi-conjugate gradient methods for CFD problems, Computational Fluid Dynamics Review, 457-476 (1995), Chichester, UK: Wiley, Chichester, UK · Zbl 0875.76431
[25] Zhang, J., Preconditioned iterative methods and finite difference schemes for convection-diffusion, Applied Mathematics and Computation, 109, 1, 11-30 (2000) · Zbl 1023.65110 · doi:10.1016/S0096-3003(99)00013-2
[26] Brandt, A., Multi-level adaptive solutions to boundary-value problems, Mathematics of Computation, 31, 138, 333-390 (1977) · Zbl 0373.65054 · doi:10.1090/S0025-5718-1977-0431719-X
[27] Hackbusch, W., On the multigrid method applied to difference equations, Computing, 20, 4, 291-306 (1978) · Zbl 0391.65045 · doi:10.1007/BF02252378
[28] Liu, C.; Liu, Z.; McCormick, S., An efficient multigrid scheme for elliptic equations with discontinuous coefficients, Communications in Applied Numerical Methods, 8, 9, 621-631 (1992) · Zbl 0759.65082 · doi:10.1002/cnm.1630080909
[29] Tzanos, C. P., Higher-order differencing method with a multigrid approach for the solution of the incompressible flow equations at high Reynolds numbers, Numerical Heat Transfer, Part B: Fundamentals, 22, 2, 179-198 (1992) · doi:10.1080/10407799208944978
[30] Othman, M.; Abdullah, A. R., An efficient multigrid Poisson solver, International Journal of Computer Mathematics, 71, 4, 541-553 (1999) · Zbl 0958.65129 · doi:10.1080/00207169908804828
[31] Gupta, M. M.; Kouatchou, J.; Zhang, J., Comparison of second- and fourth-order discretizations for multigrid Poisson solvers, Journal of Computational Physics, 132, 2, 226-232 (1997) · Zbl 0881.65120 · doi:10.1006/jcph.1996.5466
[32] Gupta, M. M.; Kouatchou, J.; Zhang, J., A compact multigrid solver for convection-diffusion equations, Journal of Computational Physics, 132, 1, 123-129 (1997) · Zbl 0881.65119 · doi:10.1006/jcph.1996.5627
[33] Zhang, J., Accelerated Multigrid High Accuracy Solution of the Convection-Diffusion Equation with High Reynolds Number, Numerical Methods for Partial Differential Equations, 13, 1, 77-92 (1997) · Zbl 0868.76063 · doi:10.1002/(SICI)1098-2426(199701)13:1<77::AID-NUM6>3.0.CO;2-J
[34] Zhang, J., Fast and high accuracy multigrid solution of the three-dimensional Poisson equation, Journal of Computational Physics, 143, 2, 449-461 (1998) · Zbl 0927.65141 · doi:10.1006/jcph.1998.5982
[35] Gupta, M. M.; Zhang, J., High accuracy multigrid solution of the 3D convection-diffusion equation, Applied Mathematics and Computation, 113, 2-3, 249-274 (2000) · Zbl 1023.65127 · doi:10.1016/S0096-3003(99)00085-5
[36] Zhang, J., Multigrid method and fourth-order compact scheme for 2D Poisson equation with unequal mesh-size discretization, Journal of Computational Physics, 179, 1, 170-179 (2002) · Zbl 1005.65137 · doi:10.1006/jcph.2002.7049
[37] Ge, Y., Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D Poisson equation, Journal of Computational Physics, 229, 18, 6381-6391 (2010) · Zbl 1197.65169 · doi:10.1016/j.jcp.2010.04.048
[38] Ge, Y.; Cao, F., Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems, Journal of Computational Physics, 230, 10, 4051-4070 (2011) · Zbl 1216.65173 · doi:10.1016/j.jcp.2011.02.027
[39] Ge, Y.; Cao, F.; Zhang, J., A transformation-free {HOC} scheme and multigrid method for solving the 3D Poisson equation on nonuniform grids, Journal of Computational Physics, 234, 199-216 (2013) · Zbl 1284.35146 · doi:10.1016/j.jcp.2012.09.034
[40] Braess, D., The contraction number of a multigrid method for solving the Poisson equation, Numerische Mathematik, 37, 3, 387-404 (1981) · Zbl 0461.65078 · doi:10.1007/BF01400317
[41] Phillips, R. E.; Schmidt, F. W., Multigrid techniques for the numerical solution of the diffusion equation, Numerical Heat Transfer, Part B: Fundamentals, 7, 3, 251-268 (1984) · Zbl 0565.65063 · doi:10.1080/01495728408961824
[42] Bank, R. E.; Benbourenane, M., The hierarchical basis multigrid method for convection-diffusion equations, Numerische Mathematik, 61, 1, 7-37 (1992) · Zbl 0757.65120 · doi:10.1007/BF01385495
[43] Gopalakrishnan, J.; Pasciak, J. E., Multigrid convergence for second order elliptic problems with smooth complex coefficients, Computer Methods Applied Mechanics and Engineering, 197, 49-50, 4411-4418 (2008) · Zbl 1194.65134 · doi:10.1016/j.cma.2008.05.018
[44] Wesseling, P., An Introduction to Multigrid Methods (1992), Chichester, Uk: John Wiley & Sons, Chichester, Uk · Zbl 0760.65092
[45] Zhang, J., On Convergence and Performance of Iterative Methods with Fourth-Order Compact Schemes, Numerical Methods for Partial Differential Equations, 14, 2, 263-280 (1998) · Zbl 0903.65080 · doi:10.1002/(SICI)1098-2426(199803)14:2<263::AID-NUM8>3.0.CO;2-M
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