Bouncing droplets on a billiard table

In a set of experiments, Couder et. al. demonstrate that an oscillating fluid bed may propagate a bouncing droplet through the guidance of the surface waves. We present a dynamical systems model, in the form of an iterative map, for a droplet on an oscillating bath. We examine the droplet bifurcation from bouncing to walking, and prescribe general requirements for the surface wave to support stable walking states. We show that in addition to walking, there is a region of large forcing that may support the chaotic bouncing of the droplet. Using the map, we then investigate the droplet trajectories for two different wave responses in a square (billiard ball) domain. We show that for waves which are quickly damped in space, the long time trajectories in a square domain are either non-periodic dense curves, or approach a quasiperiodic orbit. In contrast, for waves which extend over many wavelengths, at low forcing, trajectories tend to approach an array of circular attracting sets. As the forcing increases, the attracting sets break down and the droplet travels throughout space

Flux-splitting schemes for parabolic problems

To solve numerically boundary value problems for parabolic equations with mixed derivatives, the construction of difference schemes with prescribed quality faces essential difficulties. In parabolic problems, some possibilities are associated with the transition to a new formulation of the problem, where the fluxes (derivatives with respect to a spatial direction) are treated as unknown quantities. In this case, the original problem is rewritten in the form of a boundary value problem for the system of equations in the fluxes. This work deals with studying schemes with weights for parabolic equations written in the flux coordinates. Unconditionally stable flux locally one-dimensional schemes of the first and second order of approximation in time are constructed for parabolic equations without mixed derivatives. A peculiarity of the system of equations written in flux variables for equations with mixed derivatives is that there do exist coupled terms with time derivatives

Simulation of stellar instabilities with vastly different timescales using domain decomposition

Strange mode instabilities in the envelopes of massive stars lead to shock waves, which can oscillate on a much shorter timescale than that associated with the primary instability. The phenomenon is studied by direct numerical simulation using a, with respect to time, implicit Lagrangian scheme, which allows for the variation by several orders of magnitude of the dependent variables. The timestep for the simulation of the system is reduced appreciably by the shock oscillations and prevents its long term study. A procedure based on domain decomposition is proposed to surmount the difficulty of vastly different timescales in various regions of the stellar envelope and thus to enable the desired long term simulations. Criteria for domain decomposition are derived and the proper treatment of the resulting inner boundaries is discussed. Tests of the approach are presented and its viability is demonstrated by application to a model for the star P Cygni. In this investigation primarily the feasibility of domain decomposition for the problem considered is studied. We intend to use the results as the basis of an extension to two dimensional simulations.Comment: 15 pages, 10 figures, published in MNRA

Probing the hydrogen melting line at high pressures by dynamic compression

We investigate the capabilities of dynamic compression by intense heavy ion beams to yield information about the high pressure phases of hydrogen. Employing ab initio simulations and experimental data, a new wide range equation of state for hydrogen is constructed. The results show that the melting line up to its maximum as well as the transition from molecular fluids to fully ionized plasmas can be tested with the beam parameters soon to be available. We demonstrate that x-ray scattering can distinguish between phases and dissociation states

Symmetry-preserving discrete schemes for some heat transfer equations

Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterparts.Comment: 21 pages, 4 ps figure

Self-Similar Solutions to a Density-Dependent Reaction-Diffusion Model

In this paper, we investigated a density-dependent reaction-diffusion equation, $u_t = (u^{m})_{xx} + u - u^{m}$. This equation is known as the extension of the Fisher or Kolmogoroff-Petrovsky-Piscounoff equation which is widely used in the population dynamics, combustion theory and plasma physics. By employing the suitable transformation, this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated. Due to its simpler form, some exact self-similar solutions with the compact support have been obtained. The solutions, evolving from an initial state, converge to the usual traveling wave at a certain transition time. Hence, it is quite clear the connection between the self-similar solution and the traveling wave solution from these results. Moreover, the solutions were found in the manner that either propagates to the right or propagates to the left. Furthermore, the two solutions form a symmetric solution, expanding in both directions. The application on the spatiotemporal pattern formation in biological population has been mainly focused.Comment: 5 pages, 2 figures, accepted by Phys. Rev.

Conditional Lie-B\"acklund symmetry and reduction of evolution equations.

We suggest a generalization of the notion of invariance of a given partial differential equation with respect to Lie-B\"acklund vector field. Such generalization proves to be effective and enables us to construct principally new Ans\"atze reducing evolution-type equations to several ordinary differential equations. In the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations $u_t=u_{xx}+F(u,u_x)$ with poor Lie symmetry and obtain their exact solutions. It is shown that these solutions can not be constructed by means of the symmetry reduction procedure.Comment: 12 pages, latex, needs amssymb., to appear in the "Journal of Physics A: Mathematical and General" (1995

Regularized extremal shift in problems of stable control

We discuss a technical approach, based on the method of regularized extremal shift (RES), intended to help solve problems of stable control of uncertain dynamical systems. Our goal is to demonstrate the essence and abilities of the RES technique; for this purpose we construct feedback controller for approximate tracking a prescribed trajectory of an inaccurately observed system described by a parabolic equation. The controller is "resource-saving" in a sense that control resource spent for approximate tracking do not exceed those needed for tracking in an "ideal" situation where the current values of the input disturbance are fully observable. © 2013 IFIP International Federation for Information Processing.German Sci. Found. (DFG) Eur. Sci. Found. (ESF);Natl. Inst. Res. Comput. Sci. Control France (INRIA);DFG Research Center MATHEON;Weierstrass Institute for Applied Analysis and Stochastics (WIAS);European Patent Offic

Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure