Nonlinear Diffusion Coefficient Research Articles

Numerical identification of a nonlinear diffusion coefficient by discrete mollification

The discrete mollification method is a convolution-based filtering procedure suitable for the regularization of ill-posed problems. Combined with explicit space-marching finite difference schemes, it provides stability and convergence for a variety of coefficient identification problems in linear parabolic equations. In this paper, we extend such a technique to identify some nonlinear diffusion coefficients depending on an unknown space dependent function in one dimensional parabolic models. For the coefficient recovery process, we present detailed error estimates and to illustrate the performance of the algorithms, several numerical examples are included.

Myopic random walkers and exclusion processes: Single and multispecies

A motility mechanism based on a simple exclusion process, where the probability of movement of an agent depends on the number of unoccupied nearest-neighbor sites is considered. Such interacting agents are termed myopic. This problem is an extension of the famous blind or myopic ant in a labyrinth problem. For the interacting agent models considered here, each agent plays the role of an ant in a labyrinth, where the paths of allowed sites though the labyrinth consist of sites not occupied by other agents. We derive a nonlinear diffusion equation for the average occupancy of the discrete agent-based model for myopic agents. In contrast, interacting blind agents have a constant probability of movement to each of their nearest-neighbor sites, giving rise to a linear diffusion equation. Insight into the various terms in the nonlinear diffusion coefficient is obtained from a study of multiple subpopulations of interacting myopic agents, where an advection–diffusion equation for each subpopulation is derived, and from tracking an individual agent within the crowd, where a motility coefficient is extracted. Averaged discrete simulation data compares very well with the solution to the continuum models. We also compare the behavior of myopic and blind agents. The myopic motility mechanism is biologically motivated to emulate information an individual cell gathers from environment cues. The multispecies model developed and investigated here assists with the interpretation of experimental data involving the tracking subpopulations of cells within a total cell population.

Stochastic Approach for the Prediction of PSD in Crystallization Processes: Formulation and Comparative Assessment of Different Stochastic Models

A stochastic formulation for the description of antisolvent mediated crystal growth processes is discussed. In the proposed approach, the crystal size growth dynamics is driven by a deterministic growth factor coupled to a stochastic component. The evolution in time of the particle size distribution (PSD) is then described in terms of a Fokker−Planck equation. In this work, we investigate and assess comparatively the performance of the FPE approach to model the crystal size distribution based on different expressions for the stochastic component. In particular, we investigate the one-dimensional Fokker−Planck equation with a nonlinear diffusion coefficient to represent the crystal growth process. Validations against experimental data are presented for the NaCl−water−ethanol antisolvent crystallization system. It is shown that the stochastic model better suited to describe the experiments is given by the Geometric Brownian Motion (GBM), which gives an excellent agreement, with the experiments for a wide range of process conditions (i.e., antisolvent feed rate).

Asymptotic and Numerical Results for a Model of Solvent-Dependent Drug Diffusion through Polymeric Spheres

A model for drug diffusion from a spherical polymeric drug delivery device is considered. The model contains two key features. The first is that solvent diffuses into the polymer, which then transitions from a glassy to a rubbery state. The interface between the two states of polymer is modeled as a moving boundary, whose speed is governed by a kinetic law; the same moving boundary problem arises in the one-phase limit of a Stefan problem with kinetic undercooling. The second feature is that drug diffuses only through the rubbery region, with a nonlinear diffusion coefficient that depends on the concentration of solvent. We analyze the model using both formal asymptotics and numerical computation, the latter by applying a front-fixing scheme with a finite volume method. Previous results are extended and comparisons are made with linear models that work well under certain parameter regimes. Finally, a model for a multilayered drug delivery device is suggested, which allows for more flexible control of drug...

A variational approach to stochastic nonlinear parabolic problems

This work is concerned with an optimal control approach to stochastic nonlinear parabolic diffusion equations with monotonically increasing nonlinearity. This approach leads to sharper existence and uniqueness results under minimal growth conditions on nonlinear diffusion coefficients.

NONLINEAR SIMULATION OF TURBULENT FIELD LINES AND COMPARISON OF THEIR MEAN CROSS-FIELD DISPLACEMENT WITH THEORETICAL PREDICTION

A new method for the full nonlinear computation of turbulent field lines is presented that extends the Sums of Random Numbers Distribution method previously applied to the computation of generalized quasilinear (GQL) field lines. A study of the mean cross-field displacement confirms the theoretical prediction of a nonlinear diffusion regime, on the large scales where the mean cross-field displacement exceeds twice the perpendicular correlation length. The simulation results duplicate at all length scales the variations of the mean cross-field displacement predicted by the theoretical calculations, including the transitions from GQL supradiffusion to nonlinear diffusion. Both GQL and full nonlinear computations also reproduce, accurately, the predicted variations with turbulence level of the large-scale GQL and nonlinear diffusion coefficients, respectively.

Preconditioning Stochastic Galerkin Saddle Point Systems

Mixed finite element discretizations of deterministic second-order elliptic PDEs lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations, and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense, and the cost of a matrix vector product is nontrivial. We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed first-order system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of Gaussian random variables. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type for use with the minimal residual method (MINRES). By introducing so-called Kronecker product preconditioners, we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.

Attractor and traveling waves of a fluid with nonlinear diffusion and dispersion

This work completes the description of the method of estimating attraction domain for traveling waves for several systems of conservation laws with viscosity and capillarity effects proposed in our earlier work Thanh (2010) [26]. Precisely, we establish the global existence of traveling waves for an isentropic fluid with nonlinear diffusion and dispersion coefficients. The shock wave can be classical or nonclassical. Interestingly, we in particular show the existence of a traveling wave for a given Lax shock but rather nonclassical when the straight line connecting the two left-hand and right-hand states crosses the graph of the pressure function two more times in the middle region. Furthermore, we also discuss all the possibilities of saddle–stable, saddle–saddle, or stable–stable connections.

A non-linear degenerate equation for direct aggregation and traveling wave dynamics

The gregarious behavior of individuals of populations is an important factor in avoiding predators or for reproduction. Here, by using a random biased walk approach, we build a model which, after a transformation, takes the general form [u_{t}=[D(u)u_{x}]_{x}+g(u)] . The model involves a density-dependent non-linear diffusion coefficient [D] whose sign changes as the population density [u] increases. For negative values of [D] aggregation occurs, while dispersion occurs for positive values of [D] . We deal with a family of degenerate negative diffusion equations with logistic-like growth rate [g] . We study the one-dimensional traveling wave dynamics for these equations and illustrate our results with a couple of examples. A discussion of the ill-posedness of the partial differential equation problem is included.

PITCH-ANGLE DIFFUSION COEFFICIENTS OF CHARGED PARTICLES FROM COMPUTER SIMULATIONS

Pitch-angle diffusion is a key process in the theory of charged particle scattering by turbulent magnetic plasmas. This process is usually assumed to be diffusive and can, therefore, be described by a pitch-angle diffusion or Fokker–Planck coefficient. This parameter controls the parallel spatial diffusion coefficient as well as the parallel mean free path of charged particles. In the present paper, we determine pitch-angle diffusion coefficients from numerical computer simulations. These results are then compared with results from analytical theories. Especially, we compare the simulations with quasilinear, second-order, and weakly nonlinear diffusion coefficients. Such a comparison allows the test of previous theories and will lead to an improved understanding of the mechanism of particle scattering.

Feature Preserving Coupled Bidirectional Flow for Edge Sharpening and Image Enhancement

In the past decade there has been a growing amount of research concerning partial differential equations in image enhancement.In this paper,a feature preserving coupled bidirectional flow is presented,where an inverse diffusion with a soft edge decision is performed to sharpen edges along the normal directions to the isophote lines(edges),while a normal diffusion is done to remove noise and artifacts(jaggies)along the tangent directions on the contrary.The two converse diffusion forces are split into a coupled form to stop the cancellation between each other.To preserve image features,the nonlinear diffusion coefficients are adjusted according to the local differential geometry of image.Experimental results demonstrate that the algorithm substantially improves the subjective quality of the enhanced images.

Variationally consistent computational homogenization of transient heat flow

AbstractA framework for variationally consistent homogenization, combined with a generalized macro‐homogeneity condition, is exploited for the analysis of non‐linear transient heat conduction. Within this framework the classical approach of (model‐based) first‐order homogenization for stationary problems is extended to transient problems. Homogenization is then carried out in the spatial domain on representative volume elements (RVE), which are (in practice) introduced in quadrature points in standard fashion. Along with the classical averages, a higher order conservation quantity is obtained. An iterative FE2‐algorithm is devised for the case of non‐linear diffusion and storage coefficients, and it is applied to transient heat conduction in a strongly heterogeneous particle composite. Parametric studies are carried out, in particular with respect to the influence of the ‘internal length’ associated with the second‐order conservation quantity. Copyright © 2009 John Wiley & Sons, Ltd.

Zeroth and first-order homogenized approximations to nonlinear diffusion through block inclusions by an analytical approach

Approximate solutions to nonlinear diffusion systems are useful for many applications in computational science. When the heterogeneous nonlinear diffusion coefficient has high contrast values, an average solution given by upscaling the diffusion coefficient provides the average behavior of the fine-scale solution, which sometimes is infeasible to compute. This is also related to a problem that occurs during numerical simulations when it is necessary to coarsen meshes and an upscale coefficient is needed in order to build the data from the fine mesh to the coarse mesh. In this paper, we present a portable and computationally attractive procedure for obtaining not only the upscaled coefficient and the zeroth-order approximation of nonlinear diffusion systems, but also the first-order approximation which captures fine-scale features of the solution. These are possible by considering a correction to an approximate solution to the well known periodic cell-problem, obtained by a two-scale asymptotic expansion of the respective nonlinear diffusion equation. The correction allows one to obtain analytically the upscale diffusion coefficient, when the heterogeneous coefficient is periodic and rapidly oscillating describing inclusions in a main matrix. The approximate solutions provide a set of analytical basis functions used to construct the first-order approximation and also an estimate for the upper bound error implied in using the upscaled approximations. We demonstrate agreement with theoretical and published numerical results for the upscale coefficient, when heterogeneous coefficients are described by step-functions, as well as convergence properties of the approximations, corroborating with classical results from homogenization theory. Even though the results can be generalized, the emphasis is for conductivity functions of the form K ( x , u ( x ) ) = K s ( x ) k r ( u ( x ) ) , widely used for simulating flows in reservoirs.

Discontinuous Galerkin Finite Element Convergence for Incompressible Miscible Displacement Problems of Low Regularity

In this article we analyze the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous Galerkin method under minimal regularity assumptions. The main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem. In order to deal with the nonconformity of the method and to avoid overpenalization of jumps across interelement boundaries, the careful construction of a reflexive subspace of the space of bounded variation, which compactly embeds into $L^2(\Omega)$, and of a lifting operator, which is compatible with the nonlinear diffusion coefficient, are required. An equivalent skew-symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows us to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method. Numerical experiments underline the robustness of the proposed algorithm.

Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact

A connection is established between discrete stochastic model describing microscopic motion of fluctuating cells, and macroscopic equations describing dynamics of cellular density. Cells move towards chemical gradient (process called chemotaxis) with their shapes randomly fluctuating. Nonlinear diffusion equation is derived from microscopic dynamics in dimensions one and two using excluded volume approach. Nonlinear diffusion coefficient depends on cellular volume fraction and it is demonstrated to prevent collapse of cellular density. A very good agreement is shown between Monte Carlo simulations of the microscopic cellular Potts model and numerical solutions of the macroscopic equations for relatively large cellular volume fractions. Combination of microscopic and macroscopic models were used to simulate growth of structures similar to early vascular networks.

Spreading of a viscous liquid film over the corrugated surface and the heat and mass transfer calculations

There are a lot of industrial applications of structured packing. Distillation columns are one of the examples where the liquid flows over the corrugated surface as a thin film to provide a good mass-transfer surface between the liquid and vapor phase. The purpose of the present paper is to study the hydrodynamics and the heat-mass transfer of the liquid film spreading down the corrugated surfaces when the corrugation amplitude is comparable with Nusselt’s film thickness (the amplitude corresponds to a small texture of the structured packing). As a result, a nonlinear type diffusion equation is obtained to describe the evolution of the film thickness profile. The nonlinear diffusion coefficient is obtained for three cases: a smooth inclined plate, a corrugated plate with large ribs, and an inclined corrugated plate with small ribs. The equations are solved numerically. As a result, it has been obtained that the small texture significantly increases the rate of the film thickness evolution in comparison with a smooth plate. To obtain the nonlinear diffusion coefficient in the case of a small texture, the hydrodynamics of the film flow over an inclined corrugated surface are studied. The viscosity, inertia, and surface tension forces are taken into account. The calculations were carried out on the basis of the Navier-Stokes equations. The influence of the microcorrugations on both the heat transfer from the wall and the mass transfer through the free surface was investigated.

Diffusion-induced changes on exchangeable and organic bound copper fractions in acid soil samples enriched with copper

Transport-induced changes in the concentrations of NH 4Cl (Cu E) and Na-pyrophosphate (Cu P) extractable Cu in acid soils were studied by using diffusion cells in order to quantify Cu transport by measuring the variation of Cu E and Cu P in a half-cell with no copper added (receiving half-cell). An empirical approach based on the Freundlich equation was used to describe the relationship between Cu in the soil water extract, Cu E, and Cu P. After 30 d of diffusion, Cu E and Cu P in the receiving half-cells increased from 0.2 to 2.6% and from 4.1 to 24%, respectively, relative to their hypothetical concentrations after infinite time. Copper transport was found to be influenced by the nature of the soil and its moisture content. Diffusion was modeled in order to calculate the increase in Cu E and Cu P in the receiving half-cell, which were assumed to be in equilibrium with Cu in the soil water extract. Nonlinear retardation in the diffusive transport model was implemented by using the Freundlich equation to obtain nonlinear effective diffusion coefficients for each Cu fraction. The results of the simulations were consistent with their experimental counterparts, which suggests that the moisture content and retardation used in the model provide reasonable estimations of the increase in Cu P in acid soils.

Freeform determination of a nonlinear diffusion coefficient by a reduced adjoint system

In this article we deal with the determination of a diffusion coefficient function in a quasi-linear parabolic system. The motivation comes from a metallurgy setting. The solution method is based on the output least-squares approach (OLS) with minimization applying the adjoint equation. As the diffusion coefficient has an a-priori unknown form, we apply freeform determination in which the coefficient is defined by a linear interpolation. To avoid higher order of interpolation of the coefficient, as well as to enable the use of an adjoint system which is of the same complexity as the original partial differential equation, a reduced adjoint equation is used, applying a mapping strategy. This method is compared with a standard Levenberg-Marquardt approach.

A clutter removal method for the Doppler ultrasound signal based on a nonlinear diffusion equation

Strong clutter components produced by stationary and slow-moving tissue structures render the lower frequency part of the spectrogram useless and degrade the accuracy of clinical ultrasound indices. An adaptive method based on the nonlinear forward-and-backward diffusion equation (FAB-DE) is proposed to remove strong clutter components from the contaminated Doppler signal. The clutter signal is extracted first by the FAB-DE accurately, in which the nonlinear diffusion coefficient function of the FAB-DE locally adjusts according to signal features and the diffusion adaptively switches between forward and backward mode. The present method has been validated by simulated and realistic pulse wave Doppler signals, and compared with the conventional high pass filter and the matching pursuit method. The simulation results, including spectrogram, mean velocity error, standard deviation of mean velocity and signal-to-clutter ratio of a decontaminated signal, demonstrate that the present FAB-DE method can remove clutter sufficiently and retain more low blood components simultaneously as compared with the other two methods. Results of the realistic Doppler blood signal, including spectrogram and low-frequency part of the spectrum, support the conclusion drawn from simulation cases.

Diffusion of <sup>8</sup>Li Short-Lived Radiotracer in Li Ionic Conductors of NaTl-Type Intermetallic Compounds

Non-destructive and on-line Li diffusion experiments in Li ionic conductors are conducted using the short-lived !-emitting radiotracer of 8Li. The radiotracers produced as an energetic and pulsed ion beam from TRIAC (Tokai Radioactive Ion Accelerator Complex) are implanted into a structural defect mediated Li ionic conductor of NaTl-type intermetallic compounds ("-LiGa and "-LiIn). The experimental time spectra of the yields of !-particles are compared with simulated results and Li diffusion coefficients in the intermetallic compounds are extracted with an accuracy of ±10%. The diffusion coefficients obtained for "-LiGa with Li content of 43-54 at.% are discussed in terms of the interaction between Li-ion and the structural defects in the specimen, compared with the cases of "-LiAl and "-LiIn. The nonlinear Li-content dependency of Li diffusion coefficients for "-LiGa suggests that the Li diffusion with the Li-deficient region is obstructed by the defect complex composed of vacancies at the Li sites.