Compact mixed methods for convection/diffusion type problems
This paper presents a class of fourth-order compact finite difference technique for solving two-dimensional convection diffusion equation. The equation is recasted as a first-order mixed system, introducing a conservation and flux equations. Since flux appears explicitly in the mixed formulation, we...
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| Published in | Applied mathematics and computation Vol. 218; no. 10; pp. 5867 - 5876 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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15.01.2012
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| Abstract | This paper presents a class of fourth-order compact finite difference technique for solving two-dimensional convection diffusion equation. The equation is recasted as a first-order mixed system, introducing a conservation and flux equations. Since flux appears explicitly in the mixed formulation, we search a fourth-order compact approximation of the primary solution field and flux. Based on Taylor series expansion, the proposed compact mixed formulation generalizes the work of Carey and Spotz [G.F. Carey, W.F. Spotz, Higher-order compact mixed methods, Commun. Numer. Meth. Eng. 13 (1997)]. We show that their fourth-order formulation corresponds to a particular case of our presented scheme, and we extend their work to variable diffusion and convection coefficients. Some numerical experiments are performed to demonstrate the fourth-order effective convergence rate. |
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| AbstractList | This paper presents a class of fourth-order compact finite difference technique for solving two-dimensional convection diffusion equation. The equation is recasted as a first-order mixed system, introducing a conservation and flux equations. Since flux appears explicitly in the mixed formulation, we search a fourth-order compact approximation of the primary solution field and flux. Based on Taylor series expansion, the proposed compact mixed formulation generalizes the work of Carey and Spotz [G.F. Carey, W.F. Spotz, Higher-order compact mixed methods, Commun. Numer. Meth. Eng. 13 (1997)]. We show that their fourth-order formulation corresponds to a particular case of our presented scheme, and we extend their work to variable diffusion and convection coefficients. Some numerical experiments are performed to demonstrate the fourth-order effective convergence rate. |
| Author | Chesneau, Xavier Abide, Stéphane Zeghmati, Belkacem |
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| Cites_doi | 10.1002/fld.1650130606 10.1002/(SICI)1099-0887(199707)13:7<553::AID-CNM80>3.0.CO;2-O 10.1006/jcph.1998.5899 10.1002/(SICI)1098-2426(199603)12:2<235::AID-NUM6>3.0.CO;2-R 10.1016/0021-9991(92)90324-R 10.1016/S0377-0427(01)00504-0 10.1016/S0895-7177(00)00138-2 10.1108/09615539810206357 10.1016/j.amc.2003.12.023 10.1016/j.amc.2006.05.033 10.1006/jcph.2002.7172 10.1016/j.cam.2006.02.011 10.1006/jcph.1998.6141 10.1016/j.jcp.2009.05.038 10.1016/j.jcp.2009.01.003 10.1137/S106482759630379X 10.1016/j.jcp.2006.06.001 10.1002/nme.1620382008 |
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| References | Lele (b0005) 1992; 103 Lai, Tseng (b0085) 2007; 201 Mac Kinnon, Johnson (b0010) 1991; 13 Zhang, Zhao (b0050) 2005; 161 Spotz, Carey (b0070) 1998; 19 Sengupta, Vijay, Bhaumik (b0045) 2009; 228 Sengupta, Lakshmanan, Vijay (b0040) 2009; 228 Spotz, Carey (b0060) 1996; 12 Spotz, Carey (b0065) 1998; 8 Chu, Fan (b0025) 1998; 140 Zhao, Zhang, Corless (b0095) 2006; 182 Carey, Spotz (b0055) 1997; 13 Lai (b0080) 2002; 182 Sanyasiraju, Mishra (b0020) 2011; 9 Spotz, Carey (b0075) 1995; 38 Ge, Zhang (b0090) 2002; 143 Tian, Dai (b0015) 2007; 220 Chu, Fan (b0035) 2000; 32 Chu, Fan (b0030) 1999; 148 Spotz (10.1016/j.amc.2011.11.027_b0060) 1996; 12 Mac Kinnon (10.1016/j.amc.2011.11.027_b0010) 1991; 13 Sanyasiraju (10.1016/j.amc.2011.11.027_b0020) 2011; 9 Chu (10.1016/j.amc.2011.11.027_b0035) 2000; 32 Chu (10.1016/j.amc.2011.11.027_b0030) 1999; 148 Sengupta (10.1016/j.amc.2011.11.027_b0045) 2009; 228 Spotz (10.1016/j.amc.2011.11.027_b0070) 1998; 19 Lai (10.1016/j.amc.2011.11.027_b0085) 2007; 201 Tian (10.1016/j.amc.2011.11.027_b0015) 2007; 220 Spotz (10.1016/j.amc.2011.11.027_b0075) 1995; 38 Lai (10.1016/j.amc.2011.11.027_b0080) 2002; 182 Sengupta (10.1016/j.amc.2011.11.027_b0040) 2009; 228 Chu (10.1016/j.amc.2011.11.027_b0025) 1998; 140 Spotz (10.1016/j.amc.2011.11.027_b0065) 1998; 8 Ge (10.1016/j.amc.2011.11.027_b0090) 2002; 143 Carey (10.1016/j.amc.2011.11.027_b0055) 1997; 13 Lele (10.1016/j.amc.2011.11.027_b0005) 1992; 103 Zhang (10.1016/j.amc.2011.11.027_b0050) 2005; 161 Zhao (10.1016/j.amc.2011.11.027_b0095) 2006; 182 |
| References_xml | – volume: 201 start-page: 175 year: 2007 end-page: 181 ident: b0085 article-title: A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates publication-title: Journal of Computational and Applied Mathematics contributor: fullname: Tseng – volume: 228 start-page: 3048 year: 2009 end-page: 3071 ident: b0040 article-title: A new combined stable and dispersion relation preserving compact scheme for non-periodic problems publication-title: Journal of Computational Physics contributor: fullname: Vijay – volume: 13 start-page: 553 year: 1997 end-page: 564 ident: b0055 article-title: Higher-order compact mixed methods publication-title: Communications in Numerical Methods in Engineering contributor: fullname: Spotz – volume: 103 start-page: 16 year: 1992 end-page: 42 ident: b0005 article-title: Compact finite difference schemes with spectral-like resolution publication-title: Journal of Computational Physics contributor: fullname: Lele – volume: 182 start-page: 1454 year: 2006 end-page: 1469 ident: b0095 article-title: Convergence of the compact finite difference method for second-order elliptic equations publication-title: Applied Mathematics and Computation contributor: fullname: Corless – volume: 32 start-page: 323 year: 2000 end-page: 340 ident: b0035 article-title: A three-point sixth-order staggered combined compact difference scheme publication-title: Mathematical and Computer Modelling contributor: fullname: Fan – volume: 228 start-page: 6150 year: 2009 end-page: 6168 ident: b0045 article-title: Further improvement and analysis of CCD scheme: Dissipation discretization and de-aliasing properties publication-title: Journal of Computational Physics contributor: fullname: Bhaumik – volume: 143 start-page: 9 year: 2002 end-page: 27 ident: b0090 article-title: Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable 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compact formulation for the 3d poisson equation publication-title: Numerical Methods for Partial Differential Eqs. contributor: fullname: Carey – volume: 38 start-page: 3497 year: 1995 end-page: 3512 ident: b0075 article-title: High-order compact scheme for the steady stream-function vorticity equations publication-title: International Journal for Numerical Methods in Engineering contributor: fullname: Carey – volume: 161 start-page: 241 year: 2005 end-page: 251 ident: b0050 article-title: Truncation error and oscillation property of the combined compact difference scheme publication-title: Applied Mathematics and Computation contributor: fullname: Zhao – volume: 9 start-page: 897 year: 2011 end-page: 916 ident: b0020 publication-title: Exponential compact higher order scheme for nonlinear steady convection-diffusion Eqs. contributor: fullname: Mishra – volume: 19 start-page: 1 year: 1998 end-page: 14 ident: b0070 article-title: Iterative and parallel performance of high-order compact systems publication-title: SIAM Journal of Scientific Computing contributor: fullname: Carey – volume: 220 start-page: 952 year: 2007 end-page: 974 ident: b0015 article-title: High-order compact exponential finite difference methods for convection-diffusion type problems publication-title: Journal of Computational Physics contributor: fullname: Dai – volume: 140 start-page: 370 year: 1998 end-page: 399 ident: b0025 article-title: A Three-Point Combined Compact Difference Scheme publication-title: Journal of Computational Physics contributor: fullname: Fan – volume: 182 start-page: 337 year: 2002 end-page: 345 ident: b0080 article-title: A simple compact fourth-order poisson solver on polar geometry publication-title: Journal of Computational Physics contributor: fullname: Lai – volume: 13 start-page: 739 issue: 6 year: 1991 ident: 10.1016/j.amc.2011.11.027_b0010 article-title: Differential-equation-based representation of truncation errors for accurate numerical simulation 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fullname: Spotz – volume: 103 start-page: 16 issue: 1 year: 1992 ident: 10.1016/j.amc.2011.11.027_b0005 article-title: Compact finite difference schemes with spectral-like resolution publication-title: Journal of Computational Physics doi: 10.1016/0021-9991(92)90324-R contributor: fullname: Lele – volume: 143 start-page: 9 year: 2002 ident: 10.1016/j.amc.2011.11.027_b0090 article-title: Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients publication-title: Journal of Computational and Applied Mathematics doi: 10.1016/S0377-0427(01)00504-0 contributor: fullname: Ge – volume: 32 start-page: 323 year: 2000 ident: 10.1016/j.amc.2011.11.027_b0035 article-title: A three-point sixth-order staggered combined compact difference scheme publication-title: Mathematical and Computer Modelling doi: 10.1016/S0895-7177(00)00138-2 contributor: fullname: Chu – volume: 8 start-page: 288 year: 1998 ident: 10.1016/j.amc.2011.11.027_b0065 article-title: Formulation and experiments with high-order compact schemes for nonuniform grids publication-title: International Journal of Numerical Methods for Heat and Fluid Flow doi: 10.1108/09615539810206357 contributor: fullname: Spotz – volume: 161 start-page: 241 year: 2005 ident: 10.1016/j.amc.2011.11.027_b0050 article-title: Truncation error and oscillation property of the combined compact difference scheme publication-title: Applied Mathematics and Computation doi: 10.1016/j.amc.2003.12.023 contributor: fullname: Zhang – volume: 182 start-page: 1454 issue: 2 year: 2006 ident: 10.1016/j.amc.2011.11.027_b0095 article-title: Convergence of the compact finite difference method for second-order elliptic equations publication-title: Applied Mathematics and Computation doi: 10.1016/j.amc.2006.05.033 contributor: fullname: Zhao – volume: 182 start-page: 337 issue: 1 year: 2002 ident: 10.1016/j.amc.2011.11.027_b0080 article-title: A simple compact fourth-order poisson solver on polar geometry publication-title: Journal of Computational Physics doi: 10.1006/jcph.2002.7172 contributor: fullname: Lai – volume: 201 start-page: 175 year: 2007 ident: 10.1016/j.amc.2011.11.027_b0085 article-title: A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates publication-title: Journal of Computational and Applied Mathematics doi: 10.1016/j.cam.2006.02.011 contributor: fullname: Lai – volume: 148 start-page: 663 issue: 2 year: 1999 ident: 10.1016/j.amc.2011.11.027_b0030 article-title: A Three-Point Sixth-Order Nonuniform Combined Compact Difference Scheme publication-title: Journal of Computational Physics doi: 10.1006/jcph.1998.6141 contributor: fullname: Chu – volume: 228 start-page: 6150 issue: 17 year: 2009 ident: 10.1016/j.amc.2011.11.027_b0045 article-title: Further improvement and analysis of CCD scheme: Dissipation discretization and de-aliasing properties publication-title: Journal of Computational Physics doi: 10.1016/j.jcp.2009.05.038 contributor: fullname: Sengupta – volume: 228 start-page: 3048 issue: 8 year: 2009 ident: 10.1016/j.amc.2011.11.027_b0040 article-title: A new combined stable and dispersion relation preserving compact scheme for non-periodic problems publication-title: Journal of Computational Physics doi: 10.1016/j.jcp.2009.01.003 contributor: fullname: Sengupta – volume: 19 start-page: 1 year: 1998 ident: 10.1016/j.amc.2011.11.027_b0070 article-title: Iterative and parallel performance of high-order compact systems publication-title: SIAM Journal of Scientific Computing doi: 10.1137/S106482759630379X contributor: fullname: Spotz – volume: 220 start-page: 952 issue: 2 year: 2007 ident: 10.1016/j.amc.2011.11.027_b0015 article-title: High-order compact exponential finite difference methods for convection-diffusion type problems publication-title: Journal of Computational Physics doi: 10.1016/j.jcp.2006.06.001 contributor: fullname: Tian – volume: 38 start-page: 3497 year: 1995 ident: 10.1016/j.amc.2011.11.027_b0075 article-title: High-order compact scheme for the steady stream-function vorticity equations publication-title: International Journal for Numerical Methods in Engineering doi: 10.1002/nme.1620382008 contributor: fullname: Spotz – volume: 9 start-page: 897 year: 2011 ident: 10.1016/j.amc.2011.11.027_b0020 publication-title: Exponential compact higher order scheme for nonlinear steady convection-diffusion Eqs. contributor: fullname: Sanyasiraju |
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| SubjectTerms | Approximation Compact difference schemes Convection Convection/diffusion equation Convergence Diffusion Finite difference method Flux High accuracy Mathematical analysis Mathematical models |
| Title | Compact mixed methods for convection/diffusion type problems |
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