Numerical methods for stochastic partial differential equations with multiple scales

• Published in: Journal of computational physics Vol. 231; no. 6; pp. 2482 - 2497
• Main Authors: Abdulle, A., Pavliotis, G.A.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 20.03.2012; Elsevier
• Subjects: 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; 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
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2011.11.039

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.