A multiplicative Schwarz method with active subdomains for transient convection-diffusion problems
An efficient algorithm to find the solution of transient convection–diffusion problems with dominant convection is presented. The main idea is to follow the solution front and activate those subdomains where the solution satisfies a given threshold value. We call this novel method ‘the multiplicativ...
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| Published in: | International journal for numerical methods in biomedical engineering Vol. 26; no. 12; pp. 1573 - 1585 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Chichester, UK
John Wiley & Sons, Ltd
01.12.2010
Wiley |
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| ISSN: | 2040-7939, 2040-7947, 2040-7947 |
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| Abstract | An efficient algorithm to find the solution of transient convection–diffusion problems with dominant convection is presented. The main idea is to follow the solution front and activate those subdomains where the solution satisfies a given threshold value. We call this novel method ‘the multiplicative Schwarz method with active subdomains’, and it is motivated by the solution of a problem from activated‐carbon filters used in the automotive industry to reduce emissions. Numerical experiments show that this method is more efficient than the preconditioned conjugate gradient method with an incomplete Cholesky factorization. Copyright © 2009 John Wiley & Sons, Ltd. |
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| AbstractList | An efficient algorithm to find the solution of transient convection–diffusion problems with dominant convection is presented. The main idea is to follow the solution front and activate those subdomains where the solution satisfies a given threshold value. We call this novel method ‘the multiplicative Schwarz method with active subdomains’, and it is motivated by the solution of a problem from activated‐carbon filters used in the automotive industry to reduce emissions. Numerical experiments show that this method is more efficient than the preconditioned conjugate gradient method with an incomplete Cholesky factorization. Copyright © 2009 John Wiley & Sons, Ltd. An efficient algorithm to find the solution of transient convection-diffusion problems with dominant convection is presented. The main idea is to follow the solution front and activate those subdomains where the solution satisfies a given threshold value. We call this novel method 'the multiplicative Schwarz method with active subdomains', and it is motivated by the solution of a problem from activated-carbon filters used in the automotive industry to reduce emissions. Numerical experiments show that this method is more efficient than the preconditioned conjugate gradient method with an incomplete Cholesky factorization. |
| Author | Sandoval, M. L. Rodríguez-Ferran, A. |
| Author_xml | – sequence: 1 givenname: M. L. surname: Sandoval fullname: Sandoval, M. L. email: mlss@xanum.uam.mx organization: Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco #186, Col. Vicentina 09340 México, D.F – sequence: 2 givenname: A. surname: Rodríguez-Ferran fullname: Rodríguez-Ferran, A. organization: Laboratori de Càlcul Numèric (LaCàN), Departament de Matemàtica Aplicada III, E.T.S. d'Enginyers de Camins, Universitat Politècnica de Catalunya, Barcelona, Spain |
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| Keywords | Transient response Convection diffusion equation Matrix factorization Conjugate gradient methods Experimental study symmetric linear systems Convection convection-diffusion active subdomains Filters activated-carbon filters Activated carbon Automotive industry Diffusion equation Domain decomposition Preconditioning Cholesky factorization Heat transfer multiplicative Schwarz method |
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| References | Duff IS, Erisman AM, Reid JK. Direct Methods for Sparse Matrices (2nd edn). Oxford University Press: Oxford, 1992. Munksgaard N. Solving sparse symetric sets of linear equations by preconditioned conjugate gradients. ACM Transactions on Mathematical Software 1980; 6(2):206-219. Hackbusch W. Iterative Solution of Large Sparse Systems of Equations, vol. 95. Applied Mathematical Sciences. Springer: New York, 1994. Cai X-C, Gropp D, Keyes DE. A comparison of some domain decomposition algorithms and ILU preconditioned iterative methods for nonsymmetric elliptic problems. Numerical Linear Algebra with Applications 1994; 1(5):477-504. Xiao-Chuan C, Sarkis M. Local multiplicative Schwarz algorithms for steady and unsteady convection-diffusion equations. East-West Journal of Numerical Mathematics 1998; 6(1):27-41. Saad Y. Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics: Philadelphia, 2003. Ferragut L, Asensio MI, Monedero S. Modelling radiation and moisture content in fire spread. Communications in Numerical Methods in Engineering 2007; 23(9):819-833. Cai X-C, Saad Y. Overlapping domain decomposition algorithms for general sparse matrices. Numerical Linear Algebra with Applications 1996; 3(3):221-237. Rodríguez-Ferran A, Sandoval ML. Numerical performance of incomplete factorizations for 3D transient convection-diffusion problems. Advances in Engineering Software 2007; 38(6):439-450. Singh KM, Williams JJR. Application of the additive Schwarz method to large scale Poisson problems. Communications in Numerical Methods in Engineering 2004; 20(3):193-205. Brakkee E, Vuik C, Wesseling P. Domain decomposition for the incompressible Navier-Stokes equations: solving subdomain problems accurately and inaccurately. International Journal for Numerical Methods in Fluids 1998; 26(10):1217-1238. Huerta A, Roig B, Donea J. Time-accurate solution of stabilized convection-diffusion-reaction equations: II-accuracy analysis and examples. Communications in Numerical Methods in Engineering 2002; 18(8):575-584. Tang W-P. Generalized Schwarz splittings. SIAM Journal on Scientific Computing 1992; 13(2):573-595. Huerta A, Donea J. Time-accurate solution of stabilized convection-diffusion-reaction equations: I-time and space discretization. Communications in Numerical Methods in Engineering 2002; 18(8):565-573. Bramble JH, Pasciak JE, Wang J, Xu J. Convergence estimates for product iterative methods with applications to domain decomposition. Mathematics of Computation 1991; 57(195):1-21. Garbey M, Tromeur-Dervout D. On some Aitken-like acceleration of the Schwarz method. International Journal for Numerical Methods in Fluids 2002; 40(12):1493-1513. Meurant G. Computer Solution of Large Linear Systems. Studies in Mathematics and its Applications, vol. 28. Elsevier: Amsterdam, 1999. 1998; 26 2004; 20 2002; 18 1991; 57 2002; 40 1980; 6 2006 1992; 13 2003 1992 1994; 1 1998; 6 2007; 23 1996; 3 1994; 95 2007; 38 1988 1999 e_1_2_7_5_2 e_1_2_7_4_2 e_1_2_7_3_2 e_1_2_7_2_2 e_1_2_7_9_2 e_1_2_7_8_2 e_1_2_7_6_2 e_1_2_7_19_2 e_1_2_7_17_2 e_1_2_7_16_2 e_1_2_7_15_2 e_1_2_7_14_2 e_1_2_7_13_2 e_1_2_7_12_2 e_1_2_7_11_2 e_1_2_7_10_2 Xiao‐Chuan C (e_1_2_7_7_2) 1998; 6 Duff IS (e_1_2_7_18_2) 1992 Meurant G (e_1_2_7_20_2) 1999 |
| References_xml | – reference: Meurant G. Computer Solution of Large Linear Systems. Studies in Mathematics and its Applications, vol. 28. Elsevier: Amsterdam, 1999. – reference: Munksgaard N. Solving sparse symetric sets of linear equations by preconditioned conjugate gradients. ACM Transactions on Mathematical Software 1980; 6(2):206-219. – reference: Bramble JH, Pasciak JE, Wang J, Xu J. Convergence estimates for product iterative methods with applications to domain decomposition. Mathematics of Computation 1991; 57(195):1-21. – reference: Hackbusch W. Iterative Solution of Large Sparse Systems of Equations, vol. 95. Applied Mathematical Sciences. Springer: New York, 1994. – reference: Saad Y. Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics: Philadelphia, 2003. – reference: Xiao-Chuan C, Sarkis M. Local multiplicative Schwarz algorithms for steady and unsteady convection-diffusion equations. East-West Journal of Numerical Mathematics 1998; 6(1):27-41. – reference: Rodríguez-Ferran A, Sandoval ML. Numerical performance of incomplete factorizations for 3D transient convection-diffusion problems. Advances in Engineering Software 2007; 38(6):439-450. – reference: Garbey M, Tromeur-Dervout D. On some Aitken-like acceleration of the Schwarz method. International Journal for Numerical Methods in Fluids 2002; 40(12):1493-1513. – reference: Cai X-C, Gropp D, Keyes DE. A comparison of some domain decomposition algorithms and ILU preconditioned iterative methods for nonsymmetric elliptic problems. Numerical Linear Algebra with Applications 1994; 1(5):477-504. – reference: Singh KM, Williams JJR. Application of the additive Schwarz method to large scale Poisson problems. Communications in Numerical Methods in Engineering 2004; 20(3):193-205. – reference: Huerta A, Donea J. Time-accurate solution of stabilized convection-diffusion-reaction equations: I-time and space discretization. Communications in Numerical Methods in Engineering 2002; 18(8):565-573. – reference: Huerta A, Roig B, Donea J. Time-accurate solution of stabilized convection-diffusion-reaction equations: II-accuracy analysis and examples. Communications in Numerical Methods in Engineering 2002; 18(8):575-584. – reference: Tang W-P. Generalized Schwarz splittings. SIAM Journal on Scientific Computing 1992; 13(2):573-595. – reference: Duff IS, Erisman AM, Reid JK. Direct Methods for Sparse Matrices (2nd edn). Oxford University Press: Oxford, 1992. – reference: Ferragut L, Asensio MI, Monedero S. Modelling radiation and moisture content in fire spread. Communications in Numerical Methods in Engineering 2007; 23(9):819-833. – reference: Brakkee E, Vuik C, Wesseling P. Domain decomposition for the incompressible Navier-Stokes equations: solving subdomain problems accurately and inaccurately. International Journal for Numerical Methods in Fluids 1998; 26(10):1217-1238. – reference: Cai X-C, Saad Y. Overlapping domain decomposition algorithms for general sparse matrices. Numerical Linear Algebra with Applications 1996; 3(3):221-237. – volume: 95 year: 1994 – volume: 23 start-page: 819 issue: 9 year: 2007 end-page: 833 article-title: Modelling radiation and moisture content in fire spread publication-title: Communications in Numerical Methods in Engineering – volume: 6 start-page: 27 issue: 1 year: 1998 end-page: 41 article-title: Local multiplicative Schwarz algorithms for steady and unsteady convection–diffusion equations publication-title: East–West Journal of Numerical Mathematics – year: 1988 – year: 2006 – volume: 57 start-page: 1 issue: 195 year: 1991 end-page: 21 article-title: Convergence estimates for product iterative methods with applications to domain decomposition publication-title: Mathematics of Computation – year: 2003 – volume: 20 start-page: 193 issue: 3 year: 2004 end-page: 205 article-title: Application of the additive Schwarz method to large scale Poisson problems publication-title: Communications in Numerical Methods in Engineering – volume: 1 start-page: 477 issue: 5 year: 1994 end-page: 504 article-title: A comparison of some domain decomposition algorithms and ILU preconditioned iterative methods for nonsymmetric elliptic problems publication-title: Numerical Linear Algebra with Applications – volume: 40 start-page: 1493 issue: 12 year: 2002 end-page: 1513 article-title: On some Aitken‐like acceleration of the Schwarz method publication-title: International Journal for Numerical Methods in Fluids – volume: 18 start-page: 575 issue: 8 year: 2002 end-page: 584 article-title: Time‐accurate solution of stabilized convection–diffusion‐reaction equations: II—accuracy analysis and examples publication-title: Communications in Numerical Methods in Engineering – year: 1992 – volume: 38 start-page: 439 issue: 6 year: 2007 end-page: 450 article-title: Numerical performance of incomplete factorizations for 3D transient convection–diffusion problems publication-title: Advances in Engineering Software – volume: 13 start-page: 573 issue: 2 year: 1992 end-page: 595 article-title: Generalized Schwarz splittings publication-title: SIAM Journal on Scientific Computing – volume: 3 start-page: 221 issue: 3 year: 1996 end-page: 237 article-title: Overlapping domain decomposition algorithms for general sparse matrices publication-title: Numerical Linear Algebra with Applications – volume: 26 start-page: 1217 issue: 10 year: 1998 end-page: 1238 article-title: Domain decomposition for the incompressible Navier–Stokes equations: solving subdomain problems accurately and inaccurately publication-title: International Journal for Numerical Methods in Fluids – volume: 18 start-page: 565 issue: 8 year: 2002 end-page: 573 article-title: Time‐accurate solution of stabilized convection–diffusion‐reaction equations: I—time and space discretization publication-title: Communications in Numerical Methods in Engineering – volume: 6 start-page: 206 issue: 2 year: 1980 end-page: 219 article-title: Solving sparse symetric sets of linear equations by preconditioned conjugate gradients publication-title: ACM Transactions on Mathematical Software – year: 1999 – ident: e_1_2_7_6_2 doi: 10.1016/j.advengsoft.2006.09.003 – volume: 6 start-page: 27 issue: 1 year: 1998 ident: e_1_2_7_7_2 article-title: Local multiplicative Schwarz algorithms for steady and unsteady convection–diffusion equations publication-title: East–West Journal of Numerical Mathematics – ident: e_1_2_7_12_2 doi: 10.1137/1.9780898718003 – ident: e_1_2_7_2_2 doi: 10.1002/cnm.927 – volume-title: Computer Solution of Large Linear Systems year: 1999 ident: e_1_2_7_20_2 – ident: e_1_2_7_8_2 doi: 10.1090/S0025-5718-1991-1090464-8 – ident: e_1_2_7_19_2 doi: 10.1145/355887.355893 – ident: e_1_2_7_17_2 doi: 10.1007/978-1-4612-4288-8 – ident: e_1_2_7_15_2 doi: 10.1002/fld.407 – volume-title: Direct Methods for Sparse Matrices year: 1992 ident: e_1_2_7_18_2 – ident: e_1_2_7_3_2 doi: 10.1007/3-540-28073-1_42 – ident: e_1_2_7_4_2 doi: 10.1002/cnm.517 – ident: e_1_2_7_5_2 doi: 10.1002/cnm.518 – ident: e_1_2_7_13_2 doi: 10.1002/nla.1680010504 – ident: e_1_2_7_14_2 doi: 10.1002/(SICI)1097-0363(19980615)26:10<1217::AID-FLD693>3.0.CO;2-M – ident: e_1_2_7_10_2 doi: 10.1137/0913032 – ident: e_1_2_7_11_2 doi: 10.1002/(SICI)1099-1506(199605/06)3:3<221::AID-NLA80>3.0.CO;2-7 – ident: e_1_2_7_9_2 – ident: e_1_2_7_16_2 doi: 10.1002/cnm.660 |
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| SubjectTerms | activated-carbon filters active subdomains Algorithms Automotive components Automotive industry Cholesky factorization Conjugate gradient method Convection Convection and heat transfer convection-diffusion Emissions control Exact sciences and technology Fluid dynamics Fundamental areas of phenomenology (including applications) General theory Mathematical models multiplicative Schwarz method Physics symmetric linear systems Turbulent flows, convection, and heat transfer |
| Title | A multiplicative Schwarz method with active subdomains for transient convection-diffusion problems |
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