Nonlinear Diffusion Term Research Articles

Turing instability in reaction-diffusion systems with nonlinear diffusion

The Turing instability is studied in two-component reaction-diffusion systems with nonlinear diffusion terms, and the regions in parametric space where Turing patterns can form are determined. The boundaries between super- and subcritical bifurcations are found. Calculations are performed for one-dimensional brusselator and oregonator models.

Diffusion of two molecular species in a crowded environment: theory and experiments

Diffusion of a two component fluid is studied in the framework of differential equations, but where these equations are systematically derived from a well-defined microscopic model. The model has a finite carrying capacity imposed upon it at the mesoscopic level and this is shown to lead to nonlinear cross diffusion terms that modify the conventional Fickean picture. After reviewing the derivation of the model, the experiments carried out to test the model are described. It is found that it can adequately explain the dynamics of two dense ink drops simultaneously evolving in a container filled with water. The experiment shows that molecular crowding results in the formation of a dynamical barrier that prevents the mixing of the drops. This phenomenon is successfully captured by the model. This suggests that the proposed model can be justifiably viewed as a generalization of standard diffusion to a multispecies setting, where crowding and steric interferences are taken into account.

A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations

An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.

Numerical solutions for thin film flow down the outside and inside of a vertical cylinder

We consider a model for thin film flow down the outside and inside of a vertical cylinder. Our focus is to study the effect that the curvature of the cylinder has on the gravity-driven instability of the advancing contact line and to simulate the resulting fingering patterns that form due to this instability. The governing partial differential equation is fourth order with a nonlinear degenerate diffusion term that represents the stabilising effect of surface tension. We present numerical solutions obtained by implementing an efficient alternating direction implicit scheme. When compared to the problem of flow down a vertical plane, we find that increasing substrate curvature tends to increase the fingering instability for flow down the outside of the cylinder, whereas flow down the inside of the cylinder substrate curvature has the opposite effect. Further, we demonstrate the existence of nontrivial travelling wave solutions which describe fingering patterns that propagate down the inside of a cylinder at constant speed without changing form. These solutions are perfectly analogous to those found previously for thin film flow down an inclined plane.

Convergence of the implicit-explicit Euler scheme applied to perturbed dissipative evolution equations

We present a convergence analysis for the implicit-explicit (IMEX) Euler discretization of nonlinear evolution equations. The governing vector field of such an equation is assumed to be the sum of an unbounded dissipative operator and a Lipschitz continuous perturbation. By employing the theory of dissipative operators on Banach spaces, we prove that the IMEX Euler and the implicit Euler schemes have the same convergence order, i.e., between one half and one depending on the initial values and the vector fields. Concrete applications include the discretization of diffusion-reaction systems, with fully nonlinear and degenerate diffusion terms. The convergence and efficiency of the IMEX Euler scheme are also illustrated by a set of numerical experiments. (Less)

Adaptive numerical methods for an hydrodynamic problem arising in magnetic reading devices

The mechanical behavior of magnetic reading devices is mainly governed by compressible Reynolds equations when the air bearing modeling approximation is considered. First, the convection dominated feature motivates the use of a characteristics scheme adapted to steady state problems. Secondly, a duality method to treat the particular nonlinear diffusion term is applied. A piecewise linear finite element for spatial discretization has been chosen. Moreover, in certain conditions and devices, strong air pressure gradients arise locally, either due to a strongly convection dominated regime or to the presence of slots in the storage device, for example. In the present work we improve the previous numerical methods proposed to cope with this new setting. Thus, mainly adaptive mesh refinement algorithms based on pressure gradient indicators and appropriate multigrid techniques to solve the linear systems arising at each iteration of the duality method are proposed. Finally, several examples illustrate the performance of the set of numerical techniques.

Driven translocation of a polymer: Fluctuations at work

The impact of thermal fluctuations on the translocation dynamics of a polymer chain driven through a narrow pore has been investigated theoretically and by means of extensive Molecular-Dynamics (MD) simulation. The theoretical consideration is based on the so-called velocity Langevin (V-Langevin) equation which determines the progress of the translocation in terms of the number of polymer segments, $s(t)$, that have passed through the pore at time $t$ due to a driving force $f$. The formalism is based only on the assumption that, due to thermal fluctuations, the translocation velocity $v=\dot{s}(t)$ is a Gaussian random process as suggested by our MD data. With this in mind we have derived the corresponding Fokker-Planck equation (FPE) which has a nonlinear drift term and diffusion term with a {\em time-dependent} diffusion coefficient $D(t)$. Our MD simulation reveals that the driven translocation process follows a {\em super}diffusive law with a running diffusion coefficient $D(t) \propto t^{\gamma}$ where $\gamma < 1$. This finding is then used in the numerical solution of the FPE which yields an important result: for comparatively small driving forces fluctuations facilitate the translocation dynamics. As a consequence, the exponent $\alpha$ which describes the scaling of the mean translocation time $<\tau>$ with the length $N$ of the polymer, $<\tau> \propto N^{\alpha}$ is found to diminish. Thus, taking thermal fluctuations into account, one can explain the systematic discrepancy between theoretically predicted duration of a driven translocation process, considered usually as a deterministic event, and measurements in computer simulations. In the non-driven case, $f=0$, the translocation is slightly subdiffusive and can be treated within the framework of fractional Brownian motion (fBm).

Evolution of pattern formation under ion bombardment of substrate

Abstract Processes of self-organization described by a nonlinear evolution equation of sixth order are considered on substrate surfaces after ion beam bombardment. Five families of exact solutions are found for the nonlinear evolution equation. A numerical method enabling one to solve the boundary value problem is presented. Simulation of the pattern formation and classification of their profiles are given. The influence of the third, fifth order dispersion terms and a nonlinear diffusion term on processes of the pattern formation is discussed.

Turing Patterns in a Predator-Prey System with Self-Diffusion

For a predator-prey system, cross-diffusion has been confirmed to emerge Turing patterns. However, in the real world, the tendency for prey and predators moving along the direction of lower density of their own species, called self-diffusion, should be considered. For this, we investigate Turing instability for a predator-prey system with nonlinear diffusion terms including the normal diffusion, cross-diffusion, and self-diffusion. A sufficient condition of Turing instability for this system is obtained by analyzing the linear stability of spatial homogeneous equilibrium state of this model. A series of numerical simulations reveal Turing parameter regions of the interaction of diffusion parameters. According to these regions, we further demonstrate dispersion relations and spatial patterns. Our results indicate that self-diffusion plays an important role in the spatial patterns.

On a class of nonlocal degenerate reaction-diffusion equations with localized nonlinear diffusion term

We study a class of second order nonlocal degenerate semilinear reaction-diffusion equations with localized nonlinear diffusion term. Under a set of conditions on the localized nonlinear diffusivity and nonlinear nonlocal source term, we prove global existence and uniqueness result in the whole of some weighted Sobolev’s spaces. Furthermore, we prove nonexistence of smooth solution or blow-up of solution under some other set of conditions. Finally, we give illustrative examples for which our results apply.

Positive Steady States of a Strongly Coupled Predator-Prey System with Holling-(n+1) Functional Response

This paper discusses a predator-prey system with Holling-(n+1) functional response and the fractional type nonlinear diffusion term in a bounded domain under homogeneous Neumann boundary condition. The existence and nonexistence results concerning nonconstant positive steady states of the system were obtained. In particular, we prove that the positive constant solution(u~,v~)is asymptotically stable when the parameterksatisfies some conditions.

Pattern formation driven by cross-diffusion in a 2D domain

In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.

CROSS DIFFUSION AND NONLINEAR DIFFUSION PREVENTING BLOW UP IN THE KELLER–SEGEL MODEL

A parabolic–parabolic (Patlak–)Keller–Segel model in up to three space dimensions with nonlinear cell diffusion and an additional nonlinear cross-diffusion term is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical concentration. For arbitrarily small cross-diffusion coefficients and for suitable exponents of the nonlinear diffusion terms, the global-in-time existence of weak solutions is proved, thus preventing finite-time blow up of the cell density. The global existence result also holds for linear and fast diffusion of the cell density in a certain parameter range in three dimensions. Furthermore, we show L∞ bounds for the solutions to the parabolic–elliptic system. Sufficient conditions leading to the asymptotic stability of the constant steady state are given for a particular choice of the nonlinear diffusion exponents. Numerical experiments in two and three space dimensions illustrate the theoretical results.

Nonlinear analysis of a reaction-diffusion system: Amplitude equations

A reaction-diffusion system with a nonlinear diffusion term is considered. Based on nonlinear analysis, the amplitude equations are obtained in the cases of the Hopf and Turing instabilities in the system. Turing pattern-forming regions in the parameter space are determined for supercritical and subcritical instabilities in a two-component reaction-diffusion system.

Coexistence solutions of a periodic competition model with nonlinear diffusion

This paper is concerned with a spatially heterogeneous Lotka–Volterra competition model with nonlinear diffusion and nonlocal terms, under the Dirichlet boundary condition. Based on the theory of Leray–Schauder’s degree, we give sufficient conditions to assure the existence of coexistence periodic solutions, which extends some results of G. Fragnelli et al.

A unified term for directed and undirected motility in collective cell invasion

In this paper, we develop mathematical models for collective cell motility. Initially we develop a model using a linear diffusion–advection type equation and fit the parameters to data from cell motility assays. This approach is helpful in classifying the results of certain cell motility assay experiments. In particular, this model can determine degrees of directed versus undirected collective cell motility. Next we develop a model using a nonlinear diffusion term that is able to capture in a unified way directed and undirected collective cell motility. One goal of this work is to demonstrate that the forms of collective cell motility seen in the scratch assays and possibly other systems of interest need not reference external and more complicated migratory signals such as chemotaxis, but rather could be based on quorum sensing alone, collectively represented as density-dependent diffusivity. As an application we apply the nonlinear diffusion approach to a problem in tumor cell invasion, noting that neither chemotaxis or haptotaxis are present in the system under consideration in this article.

Pricing Climate Derivatives With Nonlinear Models

Given the large number of processes affecting global temperature in the climate system and the fact that these processes operate on different time scales, one cannot assume that a single time scale characterises either the interannual variability of climate or the response of climate to radiative forcing imposed over decades to centuries. Therefore, when modelling climate, we must consider different regimes for the time scale depending on the external forcing. We do this by introducing uncertainty to the characteristic time and to the sensitivity of climate to the stock of greenhouse gases in the atmosphere. We also make allowance for the possibility of non-linear diffusion term representing the weather and add jumps in global temperature. As a result, we improve a simple model introduced in earlier work by presenting three different jump-diffusion models for solving the estimated time constant problem and accounting for non-linear climate forcings. Copyright © 2012 Wilmott Magazine Ltd.

Fully coupled generalised hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part II: numerical scheme and numerical results

We consider the modeling and simulation of compositional two-phase flow in a porous medium, where one phase is allowed to vanish or appear. The modeling of Marchand et al. (in review) leads to a nonlinear system of two conservation equations. Each conservation equation contains several nonlinear diffusion terms, which in general cannot be written as a function of the gradients of the two principal unknowns. Also the diffusion coefficients are not necessarily explicit local functions of them. For the generalised mixed finite elements approximation, Lagrange multipliers associated to each principal unknown are introduced, the sum of the diffusive fluxes of each component is explicitly eliminated and the static condensation leads to a “global” nonlinear system of equations only in the Lagrange multipliers also including complementarity conditions to cope with vanishing or appearing phases. After time discretisation, this system can be solved at each time step using a semi-smooth Newton method. The static condensation involves “local” nonlinear systems of equations associated to each element, solved also by a semismooth Newton method. The algorithm is successfully applied to 1D and 2D examples of water–hydrogen flow involving gas phase appearance and disappearance.

On a phase field model for solid-liquid phase transitions

A new phase field model is introduced, which can be viewed as a nontrivial generalisation of what is known as the Caginalp model. It involves in particular nonlinear diffusion terms. By formal asymptotic analysis, it is shown that in the sharp interface limit it still yields a Stefan-like model with: 1) a generalized Gibbs-Thomson relation telling how much the interface temperature differs from the equilibrium temperature when the interface is moving or/and is curved with surface tension; 2) a jump condition for the heat flux, which turns out to depend on the latent heat and on the velocity of the interface with a new, nonlinear term compared to standard models. From the PDE analysis point of view, the initial-boundary value problem is proved to be locally well-posed in time (for smooth data).

A Phenomenological Approach to Modeling Transport in Porous Media

The objective of this article is to make use of the phenomenological approach to construct models for the transport of extensive quantities, such as mass of a fluid phase, mass of a component of a fluid phase, momentum of a phase and energy, in porous medium domains. Special attention is devoted to express the fluxes of these extensive quantities, especially the non-advective ones, as functions of their relevant driving forces, obeying the principle of minimum entropy production. It is shown that for each extensive quantity, we have a linear diffusive flux term, a non-linear diffusive term, and a dispersive flux term. The latter is shown to be proportional to the velocity squared. In each case, the number of moduli that describe fluid and porous matrix properties is determined. The momentum balance equation for a porous medium domain, which is the “motion equation,” is analyzed and simplified for special cases, leading to Darcy’s law and to Brinkman’s equation.