Communications in Mathematical Sciences

Volume 11 (2013)

Number 3

Subgrid-scale closure for the inviscid Burgers-Hopf equation

Pages: 757 – 777



Ulrich Achatz (Institut für Atmosphäre und Umwelt, Goethe-Universität Frankfurt, Germany)

Stamen I. Dolaptchiev (Institut für Atmosphäre und Umwelt, Goethe-Universität Frankfurt, Germany)

Ilya Timofeyev (Department of Mathematics, University of Houston, Houston, Texas, U.S.A.)


A method is presented for constructing effective stochastic models for the timeevolution of spatial averages in finite-difference discretizations of partial differential equations. This method relies on the existence of a time-scale separation in the dynamics of the spatial averages and fine-grid variables. The spatial averages, thus, are treated as the slow variables in the system and a stochastic mode reduction strategy can be applied to derive a closed form effective stochastic model. A conservative discretization of the Burgers-Hopf equation is used as a particular example to illustrate the approach. An advantage over previous applications of stochastic mode reduction to spectrally discretized models is that the resulting closure is local and thus remains applicable even if the number of slow variables is large.


stochastic parameterization, coarse-grained modeling, stochastic mode elimination

2010 Mathematics Subject Classification

60H10, 60J60, 62Mxx

Published 14 May 2013