The size of the sync basin revisited

In dynamical systems, the full stability of fixed point solutions is determined by their basins of attraction. Characterizing the structure of these basins is, in general, a complicated task, especially in high dimensionality. Recent works have advocated to quantify the non-linear stability of fixed points of dynamical systems through the relative volumes of the associated basins of attraction [Wiley et al., Chaos 16, 015103 (2006) and Menck et al. Nat. Phys. 9, 89 (2013)]. Here, we revisit this issue and propose an efficient numerical method to estimate these volumes. The algorithm first identifies stable fixed points. Second, a set of initial conditions is considered that are randomly distributed at the surface of hypercubes centered on each fixed point. These initial conditions are dynamically evolved. The linear size of each basin of attraction is finally determined by the proportion of initial conditions which converge back to the fixed point. Armed with this algorithm, we revisit the problem considered by Wiley et al. in a seminal paper [Chaos 16, 015103 (2006)] that inspired the title of the present manuscript and consider the equal-frequency Kuramoto model on a cycle. Fixed points of this model are characterized by an integer winding number q and the number n of oscillators. We find that the basin volumes scale as (1 - 4q/n)(n), contrasting with the Gaussian behavior postulated in the study by Wiley et al.. Finally, we show the applicability of our method to complex models of coupled oscillators with different natural frequencies and on meshed networks. Published by AIP Publishing.

Numerical simulation of free surface flows

Vincent Maronnier, Marco Picasso, Jacques Rappaz

A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the mathematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stokes problem are to be solved. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small rectangular cells, using a forward characteristic method. The generalized Stokes problem is solved using a finite element method on a fixed, unstructured mesh. Numerical results are presented for several test cases: the filling of an S-shaped channel, the filling of a disk with core, the broken dam in a confined domain. (C) 1999 Academic Press.

1999

Numerical homogenization methods for advection-diffusion and nonlinear monotone problems with multiple scales

Martin Ernst Huber

Many applied problems, like transport processes in porous media or ferromagnetism in composite materials, can be modeled by partial differential equations (PDEs) with heterogeneous coefficients that rapidly vary at small scales. To capture the effective behavior at a scale of interest, standard numerical methods like the finite element method (FEM) however need to resolve the finest scale of the problem (scale resolution) and thus are prohibitively costly. Numerical homogenization methods based on effective models are an efficient alternative as they require scale resolution only in a small portion of the computational domain. In this thesis, we introduce numerical homogenization methods developed in the framework of heterogeneous multiscale methods (HMM) for two different classes of multiscale PDEs. In the first part, we consider linear parabolic advection-diffusion problems with highly oscillating data, large Péclet number and compressible velocity fields. We use a discontinuous Galerkin finite element method (DG-FEM) for the spatial discretization of an effective equation whose data is estimated from finite element simulations in microscopic domains. The numerical upscaling strategy appropriately models the effects of the highly oscillating velocity field on the effective diffusion (enhanced or depleted diffusion). Thanks to the favorable stability properties of DG-FEM, robust a priori error estimates with explicit convergence rates for the macro and micro spatial discretization errors can be shown. In the second part, we propose numerical homogenization methods for nonlinear monotone multiscale problems. For parabolic problems, we first combine the backward Euler method in time with a finite element heterogeneous multiscale method (FE-HMM) in space, which couples macro and micro FEMs. We prove convergence of the method in the general

L^p(W^{1,p})

setting and provide fully discrete space-time a priori error estimates for

p=2

. The upscaling procedure however is based on nonlinear micro sampling problems, which can be computationally expensive. As a remedy, we propose a linearized method, which is only based on linear micro problems and is much more efficient as it avoids Newton iterations. We prove stability and optimal convergence rates of the linearized scheme in the classical

L^2(H^1)

setting. Further, for elliptic problems, we extend the FE-HMM (based on nonlinear micro problems) to high order macro and micro FEMs and present fully discrete a priori error estimates.

EPFL2015

Numerical homogenization methods for advection-diffusion and nonlinear monotone problems with multiple scales

Martin Ernst Huber

L^p(W^{1,p})

setting and provide fully discrete space-time a priori error estimates for

p=2

L^2(H^1)

setting. Further, for elliptic problems, we extend the FE-HMM (based on nonlinear micro problems) to high order macro and micro FEMs and present fully discrete a priori error estimates.

EPFL2015