In this paper, a family of new finite difference (NFD) methods for solving the convection-diffusion equation with singularly perturbed parameters are considered. By taking account of infinite terms in the Taylor's expansions and using the triangle function theorem, we construct a series of NFD schemes for the one-dimensional problems firstly and derive the error estimates as well. Then, applying the ADI technique, the idea is extended to two dimensional equations. Besides no numerical oscillation, there are mainly three advantages for the proposed methods: one is that the schemes can achieve the predicted convergence orders on uniform mesh regardless of the perturbed parameter for 1D equations; Secondly, no matter which convergence order the scheme is, the generated linear systems have diagonal structures; Thirdly, the methods are easily expanded to the special mesh technique such as Shishkin mesh. Some numerical experiments are shown to verify the prediction.
Taiwanese J. Math.
22(4):
949-978
(August, 2018).
DOI: 10.11650/tjm/171002
O. Axelsson and M. Nikolova, Adaptive refinement for convection-diffusion problems based on a defect-correction technique and finite difference method, Computing 58 (1997), no. 1, 1–30. 0876.35009 10.1007/BF02684469 O. Axelsson and M. Nikolova, Adaptive refinement for convection-diffusion problems based on a defect-correction technique and finite difference method, Computing 58 (1997), no. 1, 1–30. 0876.35009 10.1007/BF02684469
C. E. Baumann and J. T. Oden, A discontinuous $hp$ finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 175 (1999), no. 3-4, 311–341. 0924.76051 10.1016/S0045-7825(98)00359-4 C. E. Baumann and J. T. Oden, A discontinuous $hp$ finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 175 (1999), no. 3-4, 311–341. 0924.76051 10.1016/S0045-7825(98)00359-4
P. H. Chiu and T. W. H. Sheu, On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection-diffusion equation, J. Comput. Phys. 228 (2009), no. 10, 3640–3655. 1166.65391 10.1016/j.jcp.2009.02.008 P. H. Chiu and T. W. H. Sheu, On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection-diffusion equation, J. Comput. Phys. 228 (2009), no. 10, 3640–3655. 1166.65391 10.1016/j.jcp.2009.02.008
P. C. Chu and C. Fan, A three-point combined compact difference scheme, J. Comput. Phys. 140 (1998), no. 2, 370–399. 0923.65071 10.1006/jcph.1998.5899 P. C. Chu and C. Fan, A three-point combined compact difference scheme, J. Comput. Phys. 140 (1998), no. 2, 370–399. 0923.65071 10.1006/jcph.1998.5899
––––, A three-point sixth-order nonuniform combined compact difference scheme, J. Comput. Phys. 148 (1999), no. 2, 663–674. MR1669664 0930.65116 10.1006/jcph.1998.6141 ––––, A three-point sixth-order nonuniform combined compact difference scheme, J. Comput. Phys. 148 (1999), no. 2, 663–674. MR1669664 0930.65116 10.1006/jcph.1998.6141
R. Frank, The method of iterated defect-correction and its application to two-point boundary value problems I, Numer. Math. 25 (1976), no. 4, 409–419. 0346.65034 R. Frank, The method of iterated defect-correction and its application to two-point boundary value problems I, Numer. Math. 25 (1976), no. 4, 409–419. 0346.65034
R. Frank and C. W. Ueberhuber, Iterated defect correction for differential equations I: Theoretical results, Computing 20 (1978), no. 3, 207–228. 0401.65046 10.1007/BF02251946 R. Frank and C. W. Ueberhuber, Iterated defect correction for differential equations I: Theoretical results, Computing 20 (1978), no. 3, 207–228. 0401.65046 10.1007/BF02251946
L. Ge and J. Zhang, High accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids, J. Comput. Phys. 171 (2001), no. 2, 560–578. 0990.65117 10.1006/jcph.2001.6794 L. Ge and J. Zhang, High accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids, J. Comput. Phys. 171 (2001), no. 2, 560–578. 0990.65117 10.1006/jcph.2001.6794
M. M. Gupta, J. Kouatchou and J. Zhang, A compact multigrid solver for convection-diffusion equations, J. Comput. Phys. 132 (1997), no. 1, 123–129. 0881.65119 10.1006/jcph.1996.5627 M. M. Gupta, J. Kouatchou and J. Zhang, A compact multigrid solver for convection-diffusion equations, J. Comput. Phys. 132 (1997), no. 1, 123–129. 0881.65119 10.1006/jcph.1996.5627
P. S. Huyakorn, Solution of steady-state, convective transport equation using an upwind finite element scheme, Appl. Math. Model. 1 (1977), no. 4, 187–195. 0362.76144 10.1016/0307-904X(77)90004-X P. S. Huyakorn, Solution of steady-state, convective transport equation using an upwind finite element scheme, Appl. Math. Model. 1 (1977), no. 4, 187–195. 0362.76144 10.1016/0307-904X(77)90004-X
R. D. Lazarov, I. D. Mishev and P. S. Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal. 33 (1996), no. 1, 31–55. 0847.65075 10.1137/0733003 R. D. Lazarov, I. D. Mishev and P. S. Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal. 33 (1996), no. 1, 31–55. 0847.65075 10.1137/0733003
J. Li, Optimal uniform convergence analysis for a singularly perturbed quasilinear reaction-diffusion problem, J. Numer. Math. 12 (2004), no. 1, 39–54. 1049.65119 10.1515/1569395041172944 J. Li, Optimal uniform convergence analysis for a singularly perturbed quasilinear reaction-diffusion problem, J. Numer. Math. 12 (2004), no. 1, 39–54. 1049.65119 10.1515/1569395041172944
J. Li and Y. Chen, Uniform convergence analysis for singularly perturbed elliptic problems with parabolic layers, Numer. Math. Theory Methods Appl. 1 (2008), no. 2, 138–149. 1174.65508 J. Li and Y. Chen, Uniform convergence analysis for singularly perturbed elliptic problems with parabolic layers, Numer. Math. Theory Methods Appl. 1 (2008), no. 2, 138–149. 1174.65508
D. Liang and W. Zhao, A high-order upwind method for the convection-diffusion problem, Comput. Methods Appl. Mech. Engrg. 147 (1997), no. 1-2, 105–115. 0897.76064 10.1016/S0045-7825(97)00004-2 D. Liang and W. Zhao, A high-order upwind method for the convection-diffusion problem, Comput. Methods Appl. Mech. Engrg. 147 (1997), no. 1-2, 105–115. 0897.76064 10.1016/S0045-7825(97)00004-2
T. Linß and M. Stynes, Numerical methods on Shishkin meshes for linear convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 28, 3527–3542. 0988.76062 10.1016/S0045-7825(00)00271-1 T. Linß and M. Stynes, Numerical methods on Shishkin meshes for linear convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 28, 3527–3542. 0988.76062 10.1016/S0045-7825(00)00271-1
M. C. Natividad and M. Stynes, Richardson extrapolation for a convection-diffusion problem using a Shishkin mesh, Appl. Numer. Math. 45 (2003), no. 2-3, 315–329. 1019.65053 10.1016/S0168-9274(02)00212-X M. C. Natividad and M. Stynes, Richardson extrapolation for a convection-diffusion problem using a Shishkin mesh, Appl. Numer. Math. 45 (2003), no. 2-3, 315–329. 1019.65053 10.1016/S0168-9274(02)00212-X
A. C. R. Pillai, Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type, Internat. J. Numer. Methods Fluids 37 (2001), no. 1, 87–106. 1046.76031 10.1002/fld.167 A. C. R. Pillai, Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type, Internat. J. Numer. Methods Fluids 37 (2001), no. 1, 87–106. 1046.76031 10.1002/fld.167
H.-G. Roos, Robust numerical methods for singularly perturbed differential equations: a survey covering 2008–2012, ISRN Appl. Math. 2012 (2012), Art. ID 379547, 30 pp. H.-G. Roos, Robust numerical methods for singularly perturbed differential equations: a survey covering 2008–2012, ISRN Appl. Math. 2012 (2012), Art. ID 379547, 30 pp.
H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-diffusion-reaction and flow problems, Second edition, Springer Series in Computational Mathematics 24, Springer-Verlag, Berlin, 2008. 1155.65087 H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-diffusion-reaction and flow problems, Second edition, Springer Series in Computational Mathematics 24, Springer-Verlag, Berlin, 2008. 1155.65087
H. Sun and J. Zhang, A high-order finite difference discretization strategy based on extrapolation for convection diffusion equations, Numer. Methods Partial Differential Equations 20 (2004), no. 1, 18–32. 1038.65108 10.1002/num.10075 MR2020248 H. Sun and J. Zhang, A high-order finite difference discretization strategy based on extrapolation for convection diffusion equations, Numer. Methods Partial Differential Equations 20 (2004), no. 1, 18–32. 1038.65108 10.1002/num.10075 MR2020248
Z. F. Tian and S. Q. Dai, High-order compact exponential finite difference methods for convection-diffusion type problems, J. Comput. Phys. 220 (2007), no. 2, 952–974. 1109.65089 10.1016/j.jcp.2006.06.001 Z. F. Tian and S. Q. Dai, High-order compact exponential finite difference methods for convection-diffusion type problems, J. Comput. Phys. 220 (2007), no. 2, 952–974. 1109.65089 10.1016/j.jcp.2006.06.001
M. Vlasak and H.-G. Roos, An optimal uniform a priori error estimate for an unsteady singularly perturbed problem, Int. J. Numer. Anal. Model. 11 (2014), no. 1, 24–33. 1310.65108 M. Vlasak and H.-G. Roos, An optimal uniform a priori error estimate for an unsteady singularly perturbed problem, Int. J. Numer. Anal. Model. 11 (2014), no. 1, 24–33. 1310.65108
R. Vulanović and L. Teofanov, On the singularly perturbed semilinear reaction-diffusion problem and its numerical solution, Int. J. Numer. Anal. Model. 13 (2016), no. 1, 41–57. 1347.65129 R. Vulanović and L. Teofanov, On the singularly perturbed semilinear reaction-diffusion problem and its numerical solution, Int. J. Numer. Anal. Model. 13 (2016), no. 1, 41–57. 1347.65129
K. Wang and Y. S. Wong, Pollution-free finite difference schemes for non-homogeneous Helmholtz equation, Int. J. Numer. Anal. Model. 11 (2014), no. 4, 787–815. MR3218349 K. Wang and Y. S. Wong, Pollution-free finite difference schemes for non-homogeneous Helmholtz equation, Int. J. Numer. Anal. Model. 11 (2014), no. 4, 787–815. MR3218349
K. Wang, Y. S. Wong and J. Deng, Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates, Commun. Comput. Phys. 17 (2015), no. 3, 779–807. 06799515 K. Wang, Y. S. Wong and J. Deng, Efficient and accurate numerical solutions for Helmholtz equation in polar and spherical coordinates, Commun. Comput. Phys. 17 (2015), no. 3, 779–807. 06799515
J. Xu and L. Zikatanov, A monotone finite element scheme for convection-diffusion equations, Math. Comp. 68 (1999), no. 228, 1429–1446. 0931.65111 10.1090/S0025-5718-99-01148-5 J. Xu and L. Zikatanov, A monotone finite element scheme for convection-diffusion equations, Math. Comp. 68 (1999), no. 228, 1429–1446. 0931.65111 10.1090/S0025-5718-99-01148-5
S. Zhai and X. Feng, A new coupled high-order compact method for the three-dimensional nonlinear biharmonic equations, Int. J. Comput Math. 91 (2014), no. 10, 2307–2325. 1304.65234 10.1080/00207160.2013.877132 S. Zhai and X. Feng, A new coupled high-order compact method for the three-dimensional nonlinear biharmonic equations, Int. J. Comput Math. 91 (2014), no. 10, 2307–2325. 1304.65234 10.1080/00207160.2013.877132
S. Zhai, X. Feng and Y. He, An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation, J. Comput. Phys. 269 (2014), 138–155. 1349.65356 10.1016/j.jcp.2014.03.020 S. Zhai, X. Feng and Y. He, An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation, J. Comput. Phys. 269 (2014), 138–155. 1349.65356 10.1016/j.jcp.2014.03.020