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Title: Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation

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Journal Article: Publisher's Accepted Manuscript
Journal Name:
Physica. D, Nonlinear Phenomena
Additional Journal Information:
Journal Volume: 340; Journal Issue: C; Related Information: CHORUS Timestamp: 2017-12-05 21:42:02; Journal ID: ISSN 0167-2789
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Lu, Fei, Lin, Kevin K., and Chorin, Alexandre J.. Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation. Netherlands: N. p., 2017. Web. doi:10.1016/j.physd.2016.09.007.
Lu, Fei, Lin, Kevin K., & Chorin, Alexandre J.. Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation. Netherlands. doi:10.1016/j.physd.2016.09.007.
Lu, Fei, Lin, Kevin K., and Chorin, Alexandre J.. Wed . "Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation". Netherlands. doi:10.1016/j.physd.2016.09.007.
title = {Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation},
author = {Lu, Fei and Lin, Kevin K. and Chorin, Alexandre J.},
abstractNote = {},
doi = {10.1016/j.physd.2016.09.007},
journal = {Physica. D, Nonlinear Phenomena},
number = C,
volume = 340,
place = {Netherlands},
year = {Wed Feb 01 00:00:00 EST 2017},
month = {Wed Feb 01 00:00:00 EST 2017}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1016/j.physd.2016.09.007

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Cited by: 5works
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  • We show that the statistical properties of the large scales of the Kuramoto-Sivashinsky equation in the extended system limit can be understood in terms of the dynamical behavior of the same equation in a small finite domain. Our method relies on the description of the solutions of this equation in terms of wavelets, and allows us to model the energy transfer between small and large scales. We show that the effective equation obtained in this way can be consistently approximated by a forced Burgers equation only for scales far from the cutoff between small and large wavelengths. {copyright} {ital 1996more » The American Physical Society.}« less
  • The paper considers initial-boundary-value problems for the Kuramoto-Sivashinsky equation both with Dirichlet boundary conditions and with Navier-type boundary conditions when t>0 and x element of {omega} subset of R{sup N}, N{<=}3. Given bounded initial data, the problems in question are shown to be uniquely globally (in t>0) solvable in relevant classes of functions. Bibliography: 21 titles.
  • An algorithm is presented to integrate nonlinear partial differential equations, which is particularly useful when accurate estimation of spatial derivatives is required. It is based on an analytic approximation method, referred to as distributed approximating functionals (DAF[close quote]s), which can be used to estimate a function and a finite number of derivatives with a specified accuracy. As an application, the Kuramoto-Sivashinsky (KS) equation is integrated in polar coordinates. Its integration requires accurate estimation of spatial derivatives, particularly close to the origin. Several stationary and nonstationary solutions of the KS equation are presented, and compared with analogous states observed in themore » combustion front of a circular burner. A two-ring, nonuniform counter-rotating state has been obtained in a KS model simulation of such a burner. [copyright] [ital 1999] [ital The American Physical Society]« less
  • In problems with O(2) symmetry, the Jacobian matrix at nontrivial steady state solutions with D{sub n} symmetry always has a zero eigenvalue due to the group orbit of solutions. We consider bifurcations which occur when complex eigenvalues also cross the imaginary axis and develop a numerical method which involves the addition of a new variable, namely the velocity of solutions drifting around the group orbit, and another equation, which has the form of a phase condition for isolating one solution on the group orbit. The bifurcating branch has a particular type of spatio-temporal symmetry which can be broken in amore » further bifurcation which gives rise to modulated travelling wave solutions which drift around the group orbit. Multiple Hopf bifurcations are also considered. The methods derived are applied to the Kuramoto-Sivashinsky equation and we give results at two different bifurcations, one of which is a multiple Hopf bifurcation. Our results give insight into the numerical results of Hyman, Nicolaenko, and Zaleski. 30 refs., 2 figs., 2 tabs.« less
  • We investigate the properties of the Kuramoto-Sivashinsky equation in two spatial dimensions. We show by an explicit, numerical, coarse-graining procedure that its long-wavelength properties are described by a stochastic, partial differential equation of the Kardar-Parisi-Zhang type. From the computed parameters in our effective, stochastic equation we argue that the length and time scales over which the correlation functions cross over from linear diffusive to those of the full nonlinear equation are very large. The behavior of the three-dimensional equation is also discussed.