Instability and Collapse in Discrete Wave Equations

2005 ◽  
Vol 5 (3) ◽  
pp. 223-241
Author(s):  
A. Carpio ◽  
G. Duro
Keyword(s):  
Continuum Limit ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Blow Up ◽  
Theoretical Predictions ◽  
Initial States ◽  
The Impact ◽  
The Continuum

AbstractUnstable growth phenomena in spatially discrete wave equations are studied. We characterize sets of initial states leading to instability and collapse and obtain analytical predictions for the blow-up time. The theoretical predictions are con- trasted with the numerical solutions computed by a variety of schemes. The behavior of the systems in the continuum limit and the impact of discreteness and friction are discussed.

Numerical Solution of a Boussinesq Type Equation Using Fourier Spectral Methods

2010 ◽  
Vol 65 (4) ◽  
pp. 305-314 ◽  
Author(s):  
Hany N. Hassan
Keyword(s):  
Fourier Transform ◽  
Wave Equations ◽  
Blow Up ◽  
Type Equation ◽  
Time Discretization ◽  
Nonlinear Wave Equations ◽  
Numerical Errors ◽  
Theoretical Predictions ◽  
Space Dependence ◽  
Fourier Spectral Methods

Efficient numerical methods for solving nonlinear wave equations and studying the propagation and stability properties of their solitary waves (solitons) are applied to a Boussinesq type equation in one space dimension. These methods use a pseudospectral (Fourier transform) treatment of the space dependence, together with (a) finite differences, or (b) a fourth-order Runge-Kutta scheme (RK4), for the time evolution. Our schemes follow very accurately single solitons, which are given by simple closed formulas and are known to be stable for all allowed velocities. However, as a parameter of the problem tends to the critical value b = 0.5, where the velocity of the exact soliton vanishes, our solutions destabilize due to numerical errors, producing two small solitons in the place of the exact one. On the other hand, when we study the interaction of two such solitons, starting far apart from each other, we find in the b1, b2 parameter plane a curve beyond which the solution becomes unstable by exponential blow-up of the amplitudes, independently of our space and time discretization. We claim that this is due to a dynamical resonance rather than the accumulation of numerical errors, in agreement with theoretical predictions. Our implementation relies on the fast Fourier transform (FFT) algorithm and no major differences are observed, when either scheme (a) or (b) is used for the evolution of time.

scholarly journals A one-dimensional model for the interaction between cell-to-cell adhesion and chemotactic signalling

2011 ◽  
Vol 22 (4) ◽  
pp. 291-316 ◽  
Author(s):  
K. ANGUIGE
Keyword(s):  
Cell Adhesion ◽  
Discrete Model ◽  
Continuum Limit ◽  
Numerical Solutions ◽  
Problem Formulation ◽  
Dimensional Model ◽  
One Dimensional ◽  
Spatial Oscillations ◽  
Cell To Cell Adhesion ◽  
The Continuum

We develop and analyse a discrete, one-dimensional model of cell motility which incorporates the effects of volume filling, cell-to-cell adhesion and chemotaxis. The formal continuum limit of the model is a non-linear generalisation of the parabolic-elliptic Keller–Segel equations, with a diffusivity which can become negative if the adhesion coefficient is large. The consequent ill-posedness results in the appearance of spatial oscillations and the development of plateaus in numerical solutions of the underlying discrete model. A global-existence result is obtained for the continuum equations in the case of favourable parameter values and data, and a steady-state analysis, which, amongst other things, accounts for high-adhesion plateaus, is carried out. For ill-posed cases, a singular Stefan-problem formulation of the continuum limit is written down and solved numerically, and the numerical solutions are compared with those of the original discrete model.

EXPERIMENTAL PROPAGATION FAILURE IN A NONLINEAR ELECTRICAL LATTICE

2004 ◽  
Vol 14 (05) ◽  
pp. 1819-1830 ◽  
Author(s):  
S. BINCZAK ◽  
J. M. BILBAULT
Keyword(s):  
Analytical Solution ◽  
Traveling Waves ◽  
Continuum Limit ◽  
Experimental Results ◽  
Discrete Case ◽  
Electrical Network ◽  
Experimental Setup ◽  
Theoretical Predictions ◽  
Nagumo Equation ◽  
The Continuum

We consider an experimental setup, modeling the FitzHugh–Nagumo equation without recovery term and composed of a nonlinear electrical network made up of discrete bistable cells, resistively coupled. In the first place, we study experimentally the propagation of topological fronts in the continuum limit where the analytical solution can be obtained. We show that experimental results match the theoretical predictions. The discrete case is then investigated theoretically and in the lattice, emphasizing the pinning of traveling waves.

scholarly journals Nonlinear Kinetics on a Lattice Featuring Anomalous Diffusion

2018 ◽  
Author(s):  
Giorgio Kaniadakis ◽  
Dionissios T. Hristopulos
Keyword(s):  
Master Equation ◽  
Continuum Limit ◽  
Numerical Solutions ◽  
Planck Equation ◽  
Master Equations ◽  
Discretization Scheme ◽  
Partial Differential ◽  
Fokker Planck Equation ◽  
The Continuum ◽  
Fokker Planck

Master equations define the dynamics that govern the time evolution of various physical processes on lattices. In the continuum limit, master equations lead to Fokker-Planck partial differential equations that represent the dynamics of physical systems in continuous spaces. Over the last few decades, nonlinear Fokker-Planck equations have become very popular in condensed matter physics and in statistical physics. Numerical solutions of these equations require the use of discretization schemes. However, the discrete evolution equation obtained by the discretization of a Fokker-Planck partial differential equation depends on the specific discretization scheme. In general, the discretized form is different from the master equation that has generated the respective Fokker-Planck equation in the continuum limit. Therefore, the knowledge of the master equation associated with a given Fokker-Planck equation is extremely important for the correct numerical integration of the latter, since it provides a unique, physically motivated discretization scheme. This paper shows that the Kinetic Interaction Principle (KIP) that governs the particle kinetics of many body systems, introduced in [G. Kaniadakis, Physica A 296, 405 (2001)], univocally defines a very simple master equation that in the continuum limit yields the nonlinear Fokker-Planck equation in its most general form.

scholarly journals Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sun-Hye Park
Keyword(s):  
Time Delay ◽  
Global Existence ◽  
Wave Equations ◽  
Blow Up ◽  
Logarithmic Sobolev Inequality ◽  
Decay Rates ◽  
Infinite Time ◽  
Global Existence Of Solutions ◽  
Frictional Damping ◽  
T Tau

AbstractIn this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \alpha u_{t} (x,t) + \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$ u t t ( x , t ) − Δ u ( x , t ) + α u t ( x , t ) + β u t ( x , t − τ ) = u ( x , t ) ln | u ( x , t ) | γ . There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.

scholarly journals Spatiotemporal Correlations in the Power Output of Wind Farms: On the Impact of Atmospheric Stability

Energies ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 1486 ◽  
Author(s):  
Nicolas Tobin ◽  
Adam Lavely ◽  
Sven Schmitz ◽  
Leonardo P. Chamorro
Keyword(s):  
Power Output ◽  
Wind Farm ◽  
Statistical Significance ◽  
Atmospheric Stability ◽  
Wind Farms ◽  
Temporal Correlations ◽  
Theoretical Predictions ◽  
Power Variance ◽  
Ambient Turbulence ◽  
The Impact

The dependence of temporal correlations in the power output of wind-turbine pairs on atmospheric stability is explored using theoretical arguments and wind-farm large-eddy simulations. For this purpose, a range of five distinct stability regimes, ranging from weakly stable to moderately convective, were investigated with the same aligned wind-farm layout used among simulations. The coherence spectrum between turbine pairs in each simulation was compared to theoretical predictions. We found with high statistical significance (p < 0.01) that higher levels of atmospheric instability lead to higher coherence between turbines, with wake motions reducing correlations up to 40%. This is attributed to higher dominance of atmospheric motions over wakes in strongly unstable flows. Good agreement resulted with the use of an empirical model for wake-added turbulence to predict the variation of turbine power coherence with ambient turbulence intensity (R 2 = 0.82), though other empirical relations may be applicable. It was shown that improperly accounting for turbine–turbine correlations can substantially impact power variance estimates on the order of a factor of 4.

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