On the approximation of the stochastic Burgers equation
type: Item
instance of: scholarly article
title: On the approximation of the stochastic Burgers equation
zbMATH Open document ID: 1007.60062
author: Christoph Gugg
author: Michael Niggemann
author: Hansjörg Kielhöfer
published in: Communications in Mathematical Physics
publication date: 2002-12-02T00:00:00Z
review text: The authors consider the approximation of a stochastically perturbed hyperviscous Burgers equation \(\partial_t+u \partial_x u = (-1)^{p+1}\nu \partial_x^{2p} u +\eta\) subject to periodic boundary conditions. The random perturbation \(\eta\) comes from a large class of generalized Gaussian processes, that includes for example space-time white noise and other even rougher noises. In order to make the equation numerically accessible it is partially discretized. The main results consider an approximation of the noise by finite difference methods combined with a spectral Galerkin approximation of the equation. This gives a system of random ordinary equations, where the noise has only a finite number of degrees of freedom. The convergence of this approximation is verified in a number of different topologies. For example it is verified that any moment of the error in the uniform topology gets small. Special emphasis is given to the approximation of correlation functions like for example the \(m\)th-order velocity structure function \( E|u(t,x+r)-u(t,x)|^m.\) Possible applications of the approximation of correlation functions to fluid turbulence are discussed.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 76F30
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 65N12
zbMATH DE Number: 1839151
zbMATH Keywords: hyperviscous Burgers equation
zbMATH Keywords: spectral Galerkin method
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: correlation function
zbMATH Keywords: fluid turbulence
MaRDI profile type: Publication
full work available at URL: s00220-002-0704-5
OpenAlex ID: W1488195366
DOI: 10.1007/S00220-002-0704-5
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description: scientific article; zbMATH DE number 1839151