Numerical methods for stochastic partial differential equations with multiple scales
A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic...
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| Published in | Journal of computational physics Vol. 231; no. 6; pp. 2482 - 2497 |
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| Language | English |
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20.03.2012
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| Abstract | A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math., 58(11) (2005) 1544–1585]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blömker, M. Hairer, G.A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20(7) (2007) 1721–1744]. For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method. |
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| AbstractList | A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math., 58(11) (2005) 1544-1585]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blomker, M. Hairer, G.A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20(7) (2007) 1721-1744]. For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method. A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math., 58(11) (2005) 1544–1585]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blömker, M. Hairer, G.A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20(7) (2007) 1721–1744]. For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method. |
| Author | Pavliotis, G.A. Abdulle, A. |
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| Cites_doi | 10.1002/cpa.1014 10.1137/070679375 10.4310/CMS.2008.v6.n4.a3 10.1016/0022-1236(73)90089-X 10.1016/j.crma.2007.10.009 10.1088/0951-7715/20/7/009 10.1007/s00220-005-1368-8 10.1002/cpa.20088 10.1103/PhysRevLett.56.889 10.1090/S0025-5718-00-01224-2 10.4310/CMS.2003.v1.n2.a11 10.1103/PhysRevLett.106.060602 10.1016/0167-2789(89)90266-2 10.1051/m2an:2001148 10.1088/0951-7715/23/3/002 10.1216/RMJ-1976-6-4-653 10.1007/s00440-008-0144-z 10.1103/RevModPhys.65.851 |
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| Keywords | Homogenization Stochastic partial differential equations Heterogeneous multiscale method (HMM) Averaging Multiscale methods Spectral method Partial differential equations Digital simulation Numerical method Stochastic method Calculation methods Stochastic analysis Dynamics Differential equations Nonlinearity Calculation Stochastic equation |
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| Snippet | A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method... |
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| SubjectTerms | Amplitudes Averaging Computational techniques Exact sciences and technology Heterogeneous multiscale method (HMM) Homogenization Mathematical analysis Mathematical methods in physics Mathematical models Multiscale methods Nonlinearity Partial differential equations Physics Solvers Stochastic partial differential equations Stochasticity |
| Title | Numerical methods for stochastic partial differential equations with multiple scales |
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