Ferenc Izsák

Dispersion modeling of air pollutants in the atmosphere: a review

Modeling of dispersion of air pollutants in the atmosphere is one of the most important and challenging scientific problems. There are several natural and anthropogenic events where passive or chemically active compounds are emitted into the atmosphere. The effect of these chemical species can have serious impacts on our environment and human health. Modeling the dispersion of air pollutants can predict this effect. Therefore, development of various model strategies is a key element for the governmental and scientific communities. We provide here a brief review on the mathematical modeling of the dispersion of air pollutants in the atmosphere. We discuss the advantages and drawbacks of several model tools and strategies, namely Gaussian, Lagrangian, Eulerian and CFD models. We especially focus on several recent advances in this multidisciplinary research field, like parallel computing using graphical processing units, or adaptive mesh refinement.

Maximum likelihood estimation for constrained parameters of multinomial distributions-Application to Zipf-Mandelbrot models

A numerical maximum likelihood (ML) estimation procedure is developed for the constrained parameters of multinomial distributions. The main difficulty involved in computing the likelihood function is the precise and fast determination of the multinomial coefficients. For this the coefficients are rewritten into a telescopic product. The presented method is applied to the ML estimation of the Zipf-Mandelbrot (ZM) distribution, which provides a true model in many real-life cases. The examples discussed arise from ecological and medical observations. Based on the estimates, the hypothesis that the data is ZM distributed is tested using a chi-square test. The computer code of the presented procedure is available on request by the author.

Design of equidistant and revert type precipitation patterns in reaction–diffusion systems

In the past years considerable attention has been devoted to designing and controlling patterns at the microscale using bottom-up self-assembling techniques. The precipitation process proved itself to be a good candidate for building complex structures. Therefore, the techniques and ideas to control the precipitation processes in space and in time play an important role. We present here a simple and technologically applicable technique to produce arbitrarily shaped precipitation (Liesegang) patterns. The precipitation process is modelled using a sol coagulation model, in which the precipitation occurs if the concentration of the intermediate species (sol) produced from the initially separated reactants (inner and outer electrolytes) reaches the coagulation threshold. Spatial and/or temporal variation of this threshold can result in equidistant and revert (inverse) type patterns in contrast to regular precipitation patterns, where during the pattern formation a constant coagulation threshold is supposed and applied in the simulations. In real systems, this threshold value may be controlled by parameters which directly affect it (e.g. temperature, light intensity or ionic strength).

A new universal law for the Liesegang pattern formation

Classical regularities describing the Liesegang phenomenon have been observed and extensively studied in laboratory experiments for a long time. These have been verified in the last two decades, both theoretically and using simulations. However, they are only applicable if the observed system is driven by reaction and diffusion. We suggest here a new universal law, which is also valid in the case of various transport dynamics (purely diffusive, purely advective, and diffusion-advection cases). We state that p(tot) proportional X(c), where p(tot) yields the total amount of the precipitate and X(c) is the center of gravity. Besides the theoretical derivation experimental and numerical evidence for the universal law is provided. In contrast to the classical regularities, the introduced quantities are continuous functions of time.

Stochastic description of precipitate pattern formation in an electric field

Evolution of Liesegang patterns in an external electric field was studied numerically using a discrete stochastic model and by real experiments. In the stochastic model the diffusion was described by a Brownian random walk of discrete particles. The precipitation reaction was regarded to be a stochastic process as well, and its description was based on Ostwalds supersaturation theory. Our real experiments and the results of this stochastic approach have shown a good agreement. On the basis of our results we have proposed an extended form of the width law, which takes into account the effect of constant electric field.

Simulation of Liesegang pattern formation using a discrete stochastic model

Simulations of the Liesegang pattern formation are presented that are based on a discrete stochastic model. The diffusion term was modeled by random walk using transition probabilities referring to one and two steps. A semi-stochastic model of the precipitation process was created, using Ostwald’s supersaturation theory. The calculated variance of zone positions and formation time is in good accordance with the experimental observations of Muller et al.

Finite element approximation of fractional order elliptic boundary value problems

A finite element numerical method is investigated for fractional order elliptic boundary value problems with homogeneous Dirichlet type boundary conditions. It is pointed out that an appropriate stiffness matrix can be obtained by taking the prescribed fractional power of the stiffness matrix corresponding to the non-fractional elliptic operators. It is proved that this approach, which is also called the matrix transformation or matrix transfer method, delivers optimal rate of convergence in the L 2 -norm.

A finite difference method for fractional diffusion equations with Neumann boundary conditions

Abstract A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The wellposedness of the obtained initial value problem is proved and it is pointed out that each extension is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald–Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved.

An IMEX scheme for reaction-diffusion equations: application for a PEM fuel cell model

An implicit-explicit (IMEX) method is developed for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The corresponding method of lines scheme with finite differences is analyzed: explicit conditions are given for its convergence in the ‖·‖∞ norm. The results are applied to a model for determining the overpotential in a proton exchange membrane (PEM) fuel cell.

Error Analysis of a Continuous-Discontinuous Galerkin Finite Element Method for Generalized 2D Vorticity Dynamics

A detailed a priori error estimate is provided for a continuous-discontinuous Galerkin finite element method for the generalized two-dimensional vorticity dynamics equations. These equations describe several types of geophysical flows, including the Euler equations. The algorithm consists of a continuous Galerkin finite element method for the stream function and a discontinuous Galerkin finite element method for the (potential) vorticity. Since this algorithm satisfies a number of invariants, such as energy and enstrophy conservation, it is possible to provide detailed error estimates for this nonlinear problem. The main result of the analysis is a reduction in the smoothness requirements on the vorticity field from