ABSTRACT
We study the effects of thermal noise in a stochastic Langevin formulation of a typical example of a pattern-forming system with two-dimensional circular domain. A greater tendency towards dynamic cellular states is observed when the pattern-forming system is subjected to noise, which seems to explain the prevailing behavior of related laboratory experiments. We also report on two-dimensional numerical observations of certain dynamic states, homoclinic intermittent states, which until now, had only been observed in laboratory experiments.
ABSTRACT
We present an algorithm for the time integration of nonlinear partial differential equations. The algorithm uses distributed approximating functionals, which are based on an analytic approximation method, in order to achieve highly accurate spatial derivatives. The time integration is based on a second-order unconditionally A -stable Crank-Nicolson scheme with a Newton solver. We apply the integration scheme to the Kuramoto-Sivanshinsky equation in polar coordinates, which presents a significant computational challenge due to the stiffness introduced by the estimation of the spatial derivatives at the origin. We present several stationary and nonstationary solutions of the Kuramoto-Sivanshinsky equation and compare with previous numerical results as well as patterns observed in the combustion front of a circular burner. The numerical results of the proposed scheme reproduces several patterns--rotating two-cell, three-cell, hopping three-cell, stationary two-three-four- and five-cell, stationary 5/1,6/1,7/1,8/2 two-ring patterns, etc.--observed in physical experiments. The scheme is extremely robust and can produce long-term simulations consisting of several thousand frames. Although applied to a very specific problem, the approach of combining the framework of distributed approximating functionals with a Crank-Nicolson based time integration is generalizable to a large class of problems.
ABSTRACT
We report the first observations of numerical "hopping" cellular flame patterns found in computer simulations of the Kuramoto-Sivashinsky equation. Hopping states are characterized by nonuniform rotations of a ring of cells, in which individual cells make abrupt changes in their angular positions while they rotate around the ring. Until now, these states have been observed only in experiments but not in truly two-dimensional computer simulations. A modal decomposition analysis of the simulated patterns, via the proper orthogonal decomposition, reveals spatio-temporal behavior in which the overall temporal dynamics is similar to that of equivalent experimental states but the spatial dynamics exhibits a few more features that are not seen in the experiments. Similarities in the temporal behavior and subtle differences in the spatial dynamics between numerical hopping states and their experimental counterparts are discussed in more detail.
