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Partial semi-coarsening multigrid method based on the HOC scheme on nonuniform grids for the convection-diffusion problems. (English) Zbl 1397.65132

Authors’ abstract: A partial semi-coarsening multigrid method based on the high-order compact (HOC) difference scheme on nonuniform grids is developed to solve the 2D convection-diffusion problems with boundary or internal layers. The significance of this study is that the multigrid method allows a different number of grid points along different coordinate directions on nonuniform grids. Numerical experiments on some convection-diffusion problems with boundary or internal layers are conducted. They demonstrate that the partial semi-coarsening multigrid method combined with the HOC scheme on nonuniform grids, without losing the high-order accuracy, is very efficient and effective to decrease the computational cost by reducing the number of grid points along the direction which does not contain boundary or internal layers.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
76M20 Finite difference methods applied to problems in fluid mechanics

Software:

Wesseling
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Full Text: DOI

References:

[1] A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comput. 31 (1977), pp. 333-390. doi: 10.1090/S0025-5718-1977-0431719-X · Zbl 0373.65054
[2] G.F. Carey and W.F. Spotz, Higher-order compact mixed methods, Commun. Numer. Methods Eng. 13 (1997), pp. 553-564. doi: 10.1002/(SICI)1099-0887(199707)13:7<553::AID-CNM80>3.0.CO;2-O
[3] Q.C. Chen, Z. Gao, and Z. Yang, A perturbational h\^{}{4} exponential finite difference scheme for the convective diffusion equation, J. Comput. Phys. 104 (1993), pp. 129-139. doi: 10.1006/jcph.1993.1015 · Zbl 0774.65073
[4] R. Dai and Y. Wang, Effects of different high order compact computations for solving boundary layer problems on non-uniform grids, J. Comput. Intell. Electron. Syst. 3 (3) (2014), pp. 200-211. doi: 10.1166/jcies.2014.1091
[5] H.F. Ding and Y.X. Zhang, A new difference scheme with high accuracy and absolute stability for solving convection-diffusion equations, J. Comput. Appl. Math. 230 (2009), pp. 600-606. doi: 10.1016/j.cam.2008.12.015 · Zbl 1171.65061
[6] Y. Ge, Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D Poisson equation, J. Comput. Phys. 229 (2010), pp. 6381-6391. doi: 10.1016/j.jcp.2010.04.048 · Zbl 1197.65169
[7] Y. Ge and F. Cao, Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems, J. Comput. Phys. 230 (2011), pp. 4051-4070. doi: 10.1016/j.jcp.2011.02.027 · Zbl 1216.65173
[8] Y. Ge and F. Cao, A High Order Compact Difference Scheme and Multigrid Method for Solving the 3D Convection Diffusion Equation on Non-uniform Grids, 2012 International Conference on Computational and Information Sciences, Chongqing, August 17-19, 2012, pp. 714-717.
[9] L. Ge and J. Zhang, Accuracy, robustness, and efficiency comparison in iterative computation of convection diffusion equation with boundary layers, Numer. Methods Partial Differ. Equ. 16 (2000), pp. 379-394. doi: 10.1002/1098-2426(200007)16:4<379::AID-NUM3>3.0.CO;2-I
[10] L. Ge and J. Zhang, High accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids, J. Comput. Phys. 171 (2001), pp. 560-578. doi: 10.1006/jcph.2001.6794
[11] Y. Ge, F. Cao, and J. Zhang, A transformation-free HOC scheme and multigrid method for solving the 3D Poisson equation on nonuniform grids, J. Comput. Phys. 234 (2013), pp. 199-216. doi: 10.1016/j.jcp.2012.09.034 · Zbl 1284.35146
[12] M.M. Gupta and J. Zhang, High accuracy multigrid solution of the 3D convection-diffusion equation, Appl. Math. Comput. 113 (2000), pp. 249-274. · Zbl 1023.65127
[13] M.M. Gupta, R.P. Manohar, and J.W. Stephenson, A single cell high order scheme for the convection-diffusion equation with variable coefficients, Int. J. Numer. Methods Fluids 4 (1984), pp. 641-651. doi: 10.1002/fld.1650040704 · Zbl 0545.76096
[14] M.M. Gupta, J. Kouatchou, and J. Zhang, Comparison of second- and fourth-order discretizations for multigrid Poisson solvers, J. Comput. Phys. 132 (1997), pp. 226-232. doi: 10.1006/jcph.1996.5466
[15] M.M. Gupta, J. Kouatchou, and J. Zhang, A compact multigrid solver for convection-diffusion equations, J. Comput. Phys. 132 (1997), pp. 123-129. doi: 10.1006/jcph.1996.5627
[16] W. Hackbusch and U. Trottenberg, Multigrid Methods, Springer-Verlag, Berlin, 1982.
[17] J.C. Kalita, A.K. Dass, and D.C. Dalal, A transformation-free HOC scheme for steady convection-diffusion on non-uniform grids, Int. J. Numer. Methods Fluids 44 (2004), pp. 33-53. doi: 10.1002/fld.621 · Zbl 1062.76035
[18] S. Karaa, High-order approximation of 2D convection-diffusion equation on hexagonal grids, Numer. Methods Partial Differ. Equ. 22 (2006), pp. 1238-1246. doi: 10.1002/num.20149 · Zbl 1098.65102
[19] S. Karaa and J. Zhang, Convergence and performance of iterative methods for solving variable coefficient convection-diffusion equation with a fourth-order compact difference scheme, Comput. Math. Appl. 44 (3) (2002), pp. 457-479. doi: 10.1016/S0898-1221(02)00162-1 · Zbl 1055.65117
[20] C. Liu, Multilevel adaptive methods in computational fluid dynamics, Ph.D. thesis, University of Colorado at Denver, 1989.
[21] C. Liu and Z. Liu, Multigrid mapping and box relaxation for simulation of the whole process of flow transition in 3D boundary layers, J. Comput. Phys. 119 (1995), pp. 325-341. doi: 10.1006/jcph.1995.1138
[22] W.A. Mulder, A new multigrid approach to convection problems, J. Comput. Phys. 83 (1989), pp. 303-323. doi: 10.1016/0021-9991(89)90121-6 · Zbl 0672.76087
[23] A.C. Radhakrishna Pillai, Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type, Int. J. Numer. Methods Fluids 37 (2001), pp. 87-106. doi: 10.1002/fld.167 · Zbl 1046.76031
[24] W.F. Spotz and G.F. Carey, Formulation and experiments with high-order compact schemes for nonuniform grids, Int. J. Numer. Methods Heat Fluid Flow 8 (1998), pp. 288-303. doi: 10.1108/09615539810206357 · Zbl 0943.76060
[25] 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. 952-974. doi: 10.1016/j.jcp.2006.06.001 · Zbl 1109.65089
[26] 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. 268-286. doi: 10.1016/j.cam.2005.12.005 · Zbl 1104.65086
[27] Y. Wang and J. Zhang, Fast and robust sixth-order multigrid computation for the three-dimensional convection-diffusion equation, J. Comput. Appl. Math. 234 (2010), pp. 3496-3506. doi: 10.1016/j.cam.2010.05.022 · Zbl 1195.65149
[28] Y. Wang, S. Yu, R. Dai, and J. Zhang, A 15-point high-order compact scheme with multigrid computation for solving 3D convection diffusion equations, Int. J. Comput. Math. 92 (2) (2015), pp. 411-423. doi: 10.1080/00207160.2014.893296 · Zbl 1308.65185
[29] P. Wesseling, An Introduction to Multigrid Methods, Wiley, Chichester, 1992. · Zbl 0760.65092
[30] J. Zhang, Accelerated multigrid high accuracy solution of the convection-diffusion equation with high Reynolds number, Numer. Meth. Partial Differ. Equ. 13 (1997), pp. 77-92. doi: 10.1002/(SICI)1098-2426(199701)13:1<77::AID-NUM6>3.0.CO;2-J
[31] J. Zhang, An explicit fourth-order compact finite difference scheme for three-dimensional convection-diffusion equation, Commun. Numer. Methods Eng. 14 (1998), pp. 209-218. doi: 10.1002/(SICI)1099-0887(199803)14:3<209::AID-CNM139>3.0.CO;2-P
[32] J. Zhang, Fast and high accuracy multigrid solution of the three dimensional Poisson equation, J. Comput. Phys. 143 (1998), pp. 449-461. doi: 10.1006/jcph.1998.5982 · Zbl 0927.65141
[33] J. Zhang, On convergence and performance of iterative methods with fourth-order compact schemes, Numer. Methods Partial Differ. Equ. 14 (2) (1998), pp. 263-280. doi: 10.1002/(SICI)1098-2426(199803)14:2<263::AID-NUM8>3.0.CO;2-M
[34] J. Zhang, Multigrid method and fourth-order compact scheme for 2D Poisson equation with unequal mesh-size discretization, J. Comput. Phys. 179 (2002), pp. 170-179. doi: 10.1006/jcph.2002.7049
[35] J. Zhang, H.W. Sun, and J.J. Zhao, High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems, Comput. Methods Appl. Mech. Eng. 191 (2002), pp. 4661-4674. doi: 10.1016/S0045-7825(02)00398-5
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