4,566 research outputs found
Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation Diagrams via Stochastic Simulation
We demonstrate how time-integration of stochastic differential equations
(i.e. Brownian dynamics simulations) can be combined with continuum numerical
bifurcation analysis techniques to analyze the dynamics of liquid crystalline
polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the
approach analyzes the (unavailable in closed form) coarse macroscopic
equations, estimating the necessary quantities through appropriately
initialized, short bursts of Brownian dynamics simulation. Through this
approach, both stable and unstable branches of the equilibrium bifurcation
diagram are obtained for the Doi model of LCPs and their coarse stability is
estimated. Additional macroscopic computational tasks enabled through this
approach, such as coarse projective integration and coarse stabilizing
controller design, are also demonstrated
Geometry-induced pulse instability in microdesigned catalysts: the effect of boundary curvature
We explore the effect of boundary curvature on the instability of reactive
pulses in the catalytic oxidation of CO on microdesigned Pt catalysts. Using
ring-shaped domains of various radii, we find that the pulses disappear
(decollate from the inert boundary) at a turning point bifurcation, and trace
this boundary in both physical and geometrical parameter space. These
computations corroborate experimental observations of pulse decollation.Comment: submitted to Phys. Rev.
A multigrid continuation method for elliptic problems with folds
We introduce a new multigrid continuation method for computing solutions of nonlinear elliptic eigenvalue problems which contain limit points (also called turning points or folds). Our method combines the frozen tau technique of Brandt with pseudo-arc length continuation and correction of the parameter on the coarsest grid. This produces considerable storage savings over direct continuation methods,as well as better initial coarse grid approximations, and avoids complicated algorithms for determining the parameter on finer grids. We provide numerical results for second, fourth and sixth order approximations to the two-parameter, two-dimensional stationary reaction-diffusion problem: Δu+λ exp(u/(1+au)) = 0.
For the higher order interpolations we use bicubic and biquintic splines. The convergence rate is observed to be independent of the occurrence of limit points
Arc-Length Continuation and Multigrid Techniques for Nonlinear Elliptic Eigenvalue Problems
We investigate multi-grid methods for solving linear systems arising from arc-length continuation techniques applied to nonlinear elliptic eigenvalue problems. We find that the usual multi-grid methods diverge in the neighborhood of singular points of the solution branches. As a result, the continuation method is unable to continue past a limit point in the Bratu problem. This divergence is analyzed and a modified multi-grid algorithm has been devised based on this analysis. In principle, this new multi-grid algorithm converges for elliptic systems, arbitrarily close to singularity and has been used successfully in conjunction with arc-length continuation procedures on the model problem. In the worst situation, both the storage and the computational work are only about a factor of two more than the unmodified multi-grid methods
Stochastic Effects in Physical Systems
A tutorial review is given of some developments and applications of
stochastic processes from the point of view of the practicioner physicist. The
index is the following: 1.- Introduction 2.- Stochastic Processes 3.- Transient
Stochastic Dynamics 4.- Noise in Dynamical Systems 5.- Noise Effects in
Spatially Extended Systems 6.- Fluctuations, Phase Transitions and
Noise-Induced Transitions.Comment: 93 pages, 36 figures, LaTeX. To appear in Instabilities and
Nonequilibrium Structures VI, E. Tirapegui and W. Zeller,eds. Kluwer Academi
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