On the approximation of the stochastic Burgers equation
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.
stochastic partial differential equations
scientific article; zbMATH DE number 1839151