Abstract
In this chapter, we discuss numerical algorithms using Wiener chaos expansion (WCE) for solving second-order linear parabolic stochastic partial differential equations (SPDEs). The algorithm for computing moments of the SPDE solutions is deterministic, i.e., it does not involve any statistical errors from generating random numbers.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Matlab R2010b was used for each test on a single core of two Intel Xeon 5540 (2.53 GHz) quad-core Nehalem processors.
- 2.
This is an estimated time according to the tests with smaller Δt, L and with M = 100.
References
F.E. Benth, J. Gjerde, Convergence rates for finite element approximations of stochastic partial differential equations. Stoch. Stoch. Rep. 63, 313–326 (1998)
K. Gawȩdzki, M. Vergassola, Phase transition in the passive scalar advection. Phys. D 138, 63–90 (2000)
D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (SIAM, Philadelphia, PA, 1977)
R.H. Kraichnan, Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945–953 (1968)
H. Liu, On spectral approximations of stochastic partial differential equations driven by Poisson noise, PhD thesis, University of Southern California, 2007
S.V. LototskiÄ, B.L. RozovskiÄ, The passive scalar equation in a turbulent incompressible Gaussian velocity field. Uspekhi Mat. Nauk 59, 105–120 (2004)
S. Lototsky, R. Mikulevicius, B.L. Rozovskii, Nonlinear filtering revisited: a spectral approach. SIAM J. Control Optim. 35, 435–461 (1997)
S.V. Lototsky, B.L. Rozovskii, Wiener chaos solutions of linear stochastic evolution equations. Ann. Probab. 34, 638–662 (2006)
S.V. Lototsky, B.L. Rozovskii, Stochastic partial differential equations driven by purely spatial noise. SIAM J. Math. Anal. 41, 1295–1322 (2009)
S.V. Lototsky, K. Stemmann, Solving SPDEs driven by colored noise: a chaos approach. Quart. Appl. Math. 66, 499–520 (2008)
H. Manouzi, T.G. Theting, Mixed finite element approximation for the stochastic pressure equation of Wick type. IMA J. Numer. Anal. 24, 605–634 (2004)
R. Mikulevicius, B. Rozovskii, On unbiased stochastic Navier-Stokes equations. Probab. Theory Relat. Fields 54, 787–834 (2012)
G.N. Milstein, Y.M. Repin, M.V. Tretyakov, Numerical methods for stochastic systems preserving symplectic structure. SIAM J. Numer. Anal. 40, 1583–1604 (2002)
G.N. Milstein, M.V. Tretyakov, Stochastic Numerics for Mathematical Physics (Springer, Berlin, 2004)
G.N. Milstein, M.V. Tretyakov, Solving parabolic stochastic partial differential equations via averaging over characteristics. Math. Comp. 78, 2075–2106 (2009)
D. Nualart, B. Rozovskii, Weighted stochastic Sobolev spaces and bilinear SPDEs driven by space-time white noise. J. Funct. Anal. 149, 200–225 (1997)
B.L. RozovskiÄ, Stochastic Evolution Systems (Kluwer, Dordecht, 1990)
G. Rozza, D.B.P. Huynh, A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15, 229–275 (2008)
L.N. Trefethen, Spectral Methods in Matlab (SIAM, Philadelphia, PA, 2000)
G. Vage, Variational methods for PDEs applied to stochastic partial differential equations. Math. Scand. 82, 113–137 (1998)
D. Venturi, X. Wan, R. Mikulevicius, B.L. Rozovskii, G.E. Karniadakis, Wick-Malliavin approximation to nonlinear stochastic partial differential equations: analysis and simulations. Proc. R. Soc. Edinb. Sect. A. 469 (2013)
X. Wan, B. Rozovskii, G.E. Karniadakis, A stochastic modeling methodology based on weighted Wiener chaos and Malliavin calculus. Proc. Natl. Acad. Sci. U.S.A. 106, 14189–14194 (2009)
X. Wan, B.L. Rozovskii, The Wick–Malliavin approximation of elliptic problems with log-normal random coefficients. SIAM J. Sci. Comput. 35, A2370–A2392 (2013)
Z. Zhang, B. Rozovskii, M.V. Tretyakov, G.E. Karniadakis, A multistage Wiener chaos expansion method for stochastic advection-diffusion-reaction equations. SIAM J. Sci. Comput. 34, A914–A936 (2012)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Zhang, Z., Karniadakis, G.E. (2017). Wiener chaos methods for linear stochastic advection-diffusion-reaction equations. In: Numerical Methods for Stochastic Partial Differential Equations with White Noise. Applied Mathematical Sciences, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-319-57511-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-57511-7_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-57510-0
Online ISBN: 978-3-319-57511-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)