Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics

In this paper, we propose a numerical method based on Wiener Chaos expansion and apply it to solve the stochastic Burgers and Navier–Stokes equations driven by Brownian motion. The main advantage of the Wiener Chaos approach is that it allows for the separation of random and deterministic effects in...

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Bibliographic Details
Published in:Journal of computational physics 2006-08, Vol.216 (2), p.687-706
Main Authors: Hou, Thomas Y., Luo, Wuan, Rozovskii, Boris, Zhou, Hao-Min
Format: Article
Language:English
Subjects:
BROWNIAN MOVEMENT
CHAOS THEORY
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Computational techniques
Computer simulation
COMPUTERIZED SIMULATION
Exact sciences and technology
FLUID MECHANICS
Mathematical analysis
Mathematical methods in physics
Mathematical models
MONTE CARLO METHOD
Monte Carlo methods
NAVIER-STOKES EQUATIONS
Numerical analysis
Numerical methods
NUMERICAL SOLUTION
Physics
PROPAGATOR
RANDOMNESS
Stochastic differential equations
STOCHASTIC PROCESSES
Stochasticity
Wiener Chaos expansions
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Summary:In this paper, we propose a numerical method based on Wiener Chaos expansion and apply it to solve the stochastic Burgers and Navier–Stokes equations driven by Brownian motion. The main advantage of the Wiener Chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner. The separation principle effectively reduces a stochastic equation to its associated propagator, a system of deterministic equations for the coefficients of the Wiener Chaos expansion. Simple formulas for statistical moments of the stochastic solution are presented. These formulas only involve the solutions of the propagator. We demonstrate that for short time solutions the numerical methods based on the Wiener Chaos expansion are more efficient and accurate than those based on the Monte Carlo simulations.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2006.01.008