István Faragó

• PhD, doctor of Science
• Professor Emeritus at Eötvös Loránd University

In this paper, we extend our earlier results in Faragó, I., Karátson, J. and Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems. IMA J. Numer. Anal., 32, 1541–1573) on the discrete maximum-minimum principle (DMP) for nonlinear parabolic systems of PDEs. We propose a linearly implicit scheme, where only linear problem...

The order of accuracy of any convergent time integration method for systems of differential equations can be increased by using the sequence acceleration method known as Richardson extrapolation, as well as its variants (classical Richardson extrapolation and multiple Richardson extrapolation). The original (classical) version of Richardson extrapo...

This paper considers one dimensional unsteady heat condition in a media with temperature dependent thermal conductivity. When the thermal conductivity depends on the temperature, the corresponding heat equation is nonlinear. At one or both boundaries, a relaxing boundary condition is applied. It is a time dependent condition that approaches continu...

Richardson extrapolation is a sequence acceleration method, which has long been used to enhance the accuracy of time integration methods for solving differential equations. Its classical version is based on a suitable linear combination of numerical solutions obtained by the same numerical method with two different discretization parameters. We pre...

This article considers heat transfer in a solid body with temperature-dependent thermal conductivity that is in contact with a tank filled with liquid. The liquid in the tank is heated by hot liquid entering the tank through a pipe. Liquid at a lower temperature leaves the tank through another pipe. We propose a one-dimensional mathematical model t...

This paper addresses a reliable mathematical modeling of malaria propagation in infected societies for humans and mosquitoes with an extension of the basic Ross–Macdonald model. We analyze the extended Ross model numerically which is an initial value problem of a seven-dimensional system of the first-order ODEs. For this aim, the discretized scheme...

The consistency of the classical Richardson extrapolation (CRE), a simple and robust computational device, is analysed for the case where the underlying method is an explicit one-step numerical method for ordinary differential equations with order of consistency one or two. It is shown in the classical framework that the CRE increases the order of...

We investigate the qualitative performance of different numerical methods applied to the Ross-Macdonald malaria model. It is known that for this model a certain set is positively invariant and the question is that the discrete system which is obtained from the model by the application of a numerical method possesses this property or not. This prope...

Mathematical models are efficient tools of modelling of different phenomena. In the modelling process we formulate these phenomena in the language of mathematics. Typically, the construction of the models is realized with the modelling chain physical / biological model → continuous model → discrete (numerical) model. In order to have an adequate mo...

In mathematics there are several problems which can be described by differential equations of a certain very complicated structure. Most of the time, we cannot produce the exact (analytical) solution of these problems, so we have to approximate them numerically by using some approximating method. In this paper we analyse one of such approximation m...

In this article, a space-dependent epidemic model equipped with a constant latency period is examined. We construct a delay partial integro-differential equation and show that its solution possesses some biologically reasonable features. We propose some numerical schemes and show that, by choosing the time step to be sufficiently small, the schemes...

Atmospheric chemistry schemes, which are described mathematically by non-linear systems of ordinary differential equations (ODEs), are used in many large-scale air pollution models. These systems of ODEs are badly-scaled, extremely stiff and some components of their solution vectors vary quickly forming very sharp gradients. Therefore, it is necess...

Passive and active Richardson extrapolations are robust devices to increase the rate of convergence of time integration methods. While the order of convergence is shown to increase by one under rather natural smoothness conditions if the passive Richardson extrapolation is used, for the active Richardson extrapolation the increase of the order has...

The numerical treatment of an atmospheric chemical scheme, which contains 56 species, is discussed in this paper. This scheme is often used in studies of air pollution levels in different domains, as, for example, in Europe, by large-scale environmental models containing additionally two other important physical processes—transport of pollutants in...

In this article a space-dependent epidemic model equipped with a constant latency period is examined. We construct a delay partial integro-differential equation and show that its solution possesses some biologically reasonable features. We propose some numerical schemes and show that by choosing the time step to be sufficiently small the schemes pr...

In this investigation, operator splitting techniques have been leveraged successfully to get better accuracy of the numerical solution of a system of nonlinear ordinary differential equations representing the propagation of malaria disease as a test problem. Simulated split solutions using different operator splitting schemes, namely, the sequentia...

We consider a small object in 3D moving under the influence of a force that may depend explicitly on time, on the position of the object, and on its velocity. The equations of motion of classical mechanics are assumed to hold. If the position of the object is specified at some initial and some final time, obtaining the trajectory of the object requ...

In order to have an adequate model, the continuous and the corresponding numerical models on some fixed mesh should preserve the basic qualitative properties of the original phenomenon. In this paper we focus our attention on some mathematical models of biology, namely, we consider different epidemic models. First we investigate the SIR model, then...

Operator splitting is a powerful method for the numerical investigation of complex time-dependent models, where the stationary (elliptic) part consists of a sum of several structurally simpler sub-operators. As an alternative to the classical splitting methods, a new splitting scheme is proposed here, the Average Method with sequential splitting. I...

This is a MATLAB code that implements the numerical approach proposed in “A Numerical Approach to Solving Unsteady One-Dimensional Nonlinear Diffusion Equations”.

The diffusion problem has been widely investigated in the linear case, i.e. when the diffusion coefficient does not depend on the concentration. However, the problem with concentration-dependent diffusion coefficient, which results in nonlinear equation , is less investigated, but is also very important and is being studied actively. This work pres...

We study existence of weak solutions for certain classes of nonlinear Schr\"{o}dinger equations on the Poincar\'{e} ball model $\mathbb{B}^N$, $N\geq 3$. By using the Palais principle of symmetric criticality and suitable group theoretical arguments, we establish the existence of a nontrivial (weak) solution.

We study existence of weak solutions for certain classes of nonlinear Schrödinger equations on the Poincaré ball model BN, N≥3. By using the Palais principle of symmetric criticality and suitable group theoretical arguments, we establish the existence of a nontrivial (weak) solution.

This chapter investigates numerical solution of nonlinear two-point boundary value problems. It establishes a connection between three important, seemingly unrelated, classes of iterative methods, namely: the linearization methods, the relaxation methods (finite difference methods), and the shooting methods. It has recently been demonstrated that u...

This chapter investigates numerical solution of nonlinear two-point boundary value problems. It establishes a connection between three important, seemingly unrelated, classes of iterative methods, namely: the linearization methods, the relaxation methods (finite difference methods), and the shooting methods. It has recently been demonstrated that u...

The large-scale air pollution model UNI-DEM (the Unified Danish Eulerian Model) was used together with several carefully selected climatic scenarios. It was necessary to run the model over a long time-interval (sixteen consecutive years) and to use fine resolution on a very large space domain. This caused great difficulties because it was necessary...

In this paper, we consider the malaria propagation process for infected populations for humans and mosquitoes with an extension of the classical Ross model. The numerical model is constructed by using the semi-implicit θ-method. We show that the numerical solution and the total populations of the extended Ross model are positively invariant in spec...

The treatment of large-scale air pollution models is not only important for modern society, but also an extremely difficult task. Five important physical and chemical processes: (1) horizontal advection, (2) horizontal diffusion, (3) vertical exchange, (4) emission of different pollutants and (5) dry and wet deposition have to be united and handled...

This paper is devoted to the qualitative properties of discretized parabolic operators, such as nonnegativity and nonpositivity preservation, maximum/minimum principles and maximum norm contractivity. In the linear case, earlier papers of the authors (Faragó and Horváth in SIAM Sci Comput 28:2313–2336, 2006, IMA J Numer Anal 29:606–631, 2009) have...

We consider two classes of nonlinear eigenvalue problems with double-phase energy and lack of compactness. We establish existence and non-existence results and related properties of solutions. Our analysis combines variational methods with the generalized Pucci–Serrin maximum principle.

Complex scientific and engineering models are very useful and very significant tools in the search for answers to many difficult questions which are important for the modern society. These models are often described mathematically bynon-linearsystemsofpartialdifferentialequations.Thediscretizationofthespatial derivativesinthesesystemsleads to large...

The problem of epidemiological situation forecasting is considered. The system of differential equations that describes the development of the epidemic process with regard to spatial dependence, proposed in works I.Farago and based on the Kermak-MacKendrick model. The paper presents the results of numerical calculations for different mathematical m...

We consider a two-dimensional continuous model that describes the ecology of Easter Island. We show that the increase of the parameter corresponding to the diffusion of trees on the island has a stabilizing effect on the system, potentially preventing the collapse of island’s ecology. Next we give analytic proofs for these statements, and conduct n...

Richardson Extrapolation is a very general numerical procedure, which can be applied in the solution of many mathematical problems in an attempt to increase the accuracy of the results. It is assumed that this approach is used to handle non-linear systems of ordinary differential equations (ODEs) which arise often in the mathematical description of...

In this paper we present some extensions of the classical SIR model with non-symmetric spatial dependence. SIR-type models usually describe the epidemic in a population, which is split into three categories, namely the susceptibles (S), the infected (I) and the recovered (R). The proposed model yields a system of partial integro-differential equati...

We consider two classes of nonlinear eigenvalue problems with double-phase energy and lack of compactness. We establish existence and non-existence results and related properties of solutions. Our analysis combines variational methods with the generalized Pucci-Serrin maximum principle.

This paper studies numerical solution of nonlinear two-point boundary value problems for second order ordinary differential equations. First, it establishes a connection between the finite difference method and the quasi-linearization method. We prove that using finite differences to discretize the sequence of linear differential equations arising...

Repeated Richardson Extrapolation can successfully be used in the efforts to improve the efficiency of the numerical treatment of systems of ordinary differential equations (ODEs) mainly by increasing the accuracy of the computed results. It is assumed in this paper that Implicit Runge-Kutta Methods (IRKMs) are used in the numerical solution of sys...

Efficient implementation of the Two-times Repeated Richardson Extrapolation is studied in this paper under the assumption that systems of ordinary differential equations (ODEs) are solved numerically by Explicit Runge-Kutta Methods (ERKMs). The combinations of the Two-times Repeated Richardson Extrapolation with the ERKMs are new numerical methods....

Abstract⎯ Richardson extrapolation is a numerical procedure which enables us to enhance the accuracy of any convergent numerical method in a simple and powerful way. In this paper we overview the theoretical background of Richardson extrapolation in space and time, where two numerical solutions, obtained on a coarse and a fine spacetime grid are co...

This book explores mathematics in a wide variety of applications, ranging from problems in electronics, energy and the environment, to mechanics and mechatronics. The book gathers 81 contributions submitted to the 20th European Conference on Mathematics for Industry, ECMI 2018, which was held in Budapest, Hungary in June 2018. The application areas...

212 Int. J. Environment and Pollution, Vol. 66, Nos. 1/2/3, 2019 Copyright © 2019 Inderscience Enterprises Ltd. Advanced algorithms for studying the impact of climate changes on ozone levels in the atmosphere Zahari Zlatev Department of Environmental Science, Aarhus University, Roskilde, Denmark Email: zz@envs.au.dk Ivan Dimov Institute of In...

212 Int. J. Environment and Pollution, Vol. 66, Nos. 1/2/3, 2019 Copyright © 2019 Inderscience Enterprises Ltd. Advanced algorithms for studying the impact of climate changes on ozone levels in the atmosphere Zahari Zlatev Department of Environmental Science, Aarhus University, Roskilde, Denmark Email: zz@envs.au.dk Ivan Dimov Institute of In...

This paper considers one-dimensional heat transfer in a media with temperature-dependent thermal conductivity. To model the transient behavior of the system, we solve numerically the one-dimensional unsteady heat conduction equation with certain initial and boundary conditions. Contrary to the traditional approach, when the equation is first discre...

The solution of a parabolic problem is expected to reproduce the basic qualitative properties of the original phenomenon, such as nonnegativity/nonpositivity preservation, maximum/minimum principles and maximum norm contractivity, without which the model might lead to unrealistic quantities in conflict with physical reality. This paper presents cha...

We investigate biological processes, particularly the propagation of malaria. Both the continuous and the numerical models on some fixed mesh should preserve the basic qualitative properties of the original phenomenon. Our main goal is to give the conditions for the discrete (numerical) models of the malaria phenomena under which they possess some...

Geometric integrators are numerical methods for differential equations that preserve geometric properties. In this article we investigate the questions of constructing such methods for the well-known Lotka–Volterra predator-prey system by using the operator splitting method. We use different numerical methods combined with the operator splitting me...

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950's, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using...

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950's, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using...

In this paper we investigate discrete models which describe several important biological phenomena. Particularly, we analyse in detail the discrete models for the epidemic propagation and also for the Lotka-Volterra models. We give such conditions under which the models are qualitatively adequate, which means, that they preserve the most important,...

In this paper the qualitative properties of certain spatial disease propagation models are investigated. The paper can be considered as a generalization of the papers (Faragóand Horváth, 2016; 2017). The models of these papers assume that the members of the population do not move and that the infection is localized in the sense that only the member...

In this work the alternant theta-method and its application is investigated. We analyze the local approximation error and the convergence of the method on the non-equidistant mesh. We define the order of convergence, as well. The main idea of this approach is the approximation of the solution of the Cauchy problems by using different numerical sche...

This book constitutes thoroughly revised selected papers of the 6th International Conference on Numerical Analysis and Its Applications, NAA 2016, held in Lozenetz, Bulgaria, in June 2016. The 90 revised papers presented were carefully reviewed and selected from 98 submissions. The conference offers a wide range of the following topics: Numerical M...

It is a natural expectation that the mathematical models of real-life phenomena have to possess some characteristic qualitative properties of the original process. For parabolic problems the main known qualitative properties are the maximum-minimum principles, nonnegativity-nonpositivity preservation and maximum norm contractivity. These properties...

Discrete nonnegativity principles are established for finite element approximations of nonlinear parabolic PDE systems with mixed boundary conditions. Previous results of the authors are extended such that diagonal dominance (or essentially monotonicity) of the nonlinear coupling can be relaxed, allowing to include much more general situations in s...

Most of the models of epidemic propagations do not take into the account the spatial distribution of the individuals. They give only the temporal change of the number of the infected, susceptible and recovered patients. In our presentation we present a spatial epidemic propagation model and give some of its qualitative properties both in the contin...

In this work operator splitting techniques have been applied successfully to improve the accuracy of multi-scale Lithium-ion (Li-ion) battery models. A slightly simplified Li-ion battery model is derived, which can be solved on one time scale and multiple time scales. Different operator splitting schemes combined with different approximations are c...

The implementation of the Richardson Extrapolation in combination with different numerical methods for solving systems of ordinary differential equations (ODEs) is relatively simple, but the important requirement for stability of the computational process may cause serious difficulties. For example, the commonly used by scientists and engineers Tra...

The climatic changes are already causing more and more extreme weather events in many different areas of the Earth. One of the most important consequences of these changes, which has different impacts, is the clear trend for global increase of the temperature. This increase has also some influence on the pollution levels, mainly because (a) many of...

This book deals with mathematical problems arising in the context of meteorological modelling. It gathers and presents some of the most interesting and important issues from the interaction of mathematics and meteorology. It is unique in that it features contributions on topics like data assimilation, ensemble prediction, numerical methods, and tra...

The paper deals with discretisation methods for nonlinear operator equations written as abstract nonlinear evolution equations. Brezis and Pazy showed that the solution of such problems is given by nonlinear semigroups whose theory was founded by Crandall and Liggett. By using the approximation theorem of Brezis and Pazy, we show the -stability of...

The paper serves as a review on the basic results showing how functional analytic tools have been applied in numerical analysis. It deals with abstract Cauchy problems and present how their solutions are approximated by using space and time discretisations. To this end we introduce and apply the basic notions of operator semigroup theory. The conve...

This paper considers mathematical and numerical models of normal and abnormal tissue growth. To this aim, we analyse a nonlinear system of partial differential equations, called as cross-diffusion system, in a rather general form. We investigate the dynamics of the system in dependence on the system parameters. We show analytically the existence of...

Most of the models of epidemic propagations do not take into account the spatial distribution of the individuals. They give only the temporal change of the number of the infected, susceptible and recovered patients. In this paper we give some spatial discrete one-step iteration models for disease propagation and give conditions that guarantee some...

The invasive species model describes the connections between three species: People, trees and rats. In 2008, Basener, Brooks, Radin and Wiandt presented an article in that, they created a mathematical model for such dynamical system. In this work we changed the model and investigated the equilibrium points and stability of the invasive species mode...

This book constitutes the thoroughly refereed post-conference proceedings of the 6th International Conference on Finite Difference Methods, FDM 2014, held in Lozenetz, Bulgaria, in June 2014. The 36 revised full papers were carefully reviewed and selected from 62 submissions. These papers together with 12 invited papers cover topics such as finite...

A Crank–Nicolson type scheme, which is of order two with respect to all independent variables, is used in the numerical solution of multi-dimensional advection equations. Normally, the order of accuracy of any numerical scheme can be increased by one when the well-known Richardson Extrapolation is used. It is proved that in this particular case the...

The stability is one of the most basic requirement for the numerical model, which is mostly elaborated for the linear problems. In this paper we analyze the stability notions for the nonlinear problems. We show that, in case of consistency, both the N-stability and K-stability notions guarantee the convergence. Moreover, by using the N-stability we...

The papers in this Special Issue on 'Efficient Algorithms for Large Scale Scientific Computations' are mainly devoted to the efficient solutions related to the selection of numerical algorithms. The selected papers deal mainly with the important topics, such as, finite element methods applied in the discretization of several types of PDEs or system...

Large-scale environmental models can successfully be used in different important for the modern society studies as, for example, in the investigation of the influence of the future climatic changes on pollution levels in different countries. Such models are normally described mathematically by non-linear systems of partial differential equations, w...

In this paper we investigate the N-stability notion in an abstract Banach space setting. The main result is that thanks to this notion we have an alternative opportunity for verifying the stability of the numerical solution for periodic initial-value reaction-diffusion problems.

In this paper we investigate the T-stability of one-step methods for initial-value problems. The main result is that we extend the classical result (the well-known Euler method) for variable step size explicit and implicit one-step methods. In addition, we give further properties for the theory of T-stability of nonlinear equations in an abstract (...

The properties of iterative splitting with two bounded linear operators have been analyzed by Faragó et al. For more than two operators, iterative splitting can be defined in many different ways. A large class of the possible extensions to this case is presented in this paper and the order of accuracy of these methods are examined. A separate secti...

In this paper we investigate the N-stability notion in an abstract Banach space setting. The main result is that we verify the N-stability of the implicit difference method for the periodic initial-value reaction-diffusion problem.

An implicit-explicit (IMEX) method is combined with some so-called Richardson extrapolation (RiEx) methods for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The results are applied to a model for determining the overpotential in a Proton Exchange Membrane (PEM) fuel cell.

In this work, we present and discuss some continuous and dis-crete maximum principles for linear elliptic problem of the second order with the third boundary condition (almost never addressed to in the available literature in this context) solved by the finite element and finite difference methods. Numerical tests are given.

Large-scale mathematical models are extensively used in the modern computer age to handle many complex problems which arise in different fields of science and engineering. These models are typically described by time-dependent systems of partial differential equations (PDE). The discretization of the spatial derivatives in these systems of PDEs lea...

Runge-Kutta methods are widely used in the solution of systems of ordinary differential equations. Richardson extrapolation is an efficient tool for enhancing the accuracy of time integration schemes. In this paper we investigate the convergence of the combination of any of the diagonally implicit (including also the explicit) Runge-Kutta methods w...

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 over...

For the solution of the Cauchy problem for the first order ODE, the most popular, simplest and widely used method are the Euler methods. The two basic variants of the Euler methods are the explicit Euler methods (EEM) and the implicit Euler method (IEM). These methods are well-known and they are introduced almost in any arbitrary textbook of the nu...

Discrete maximum principles (DMPs) are established for finite element approximations of systems of nonlinear parabolic partial differential equations with mixed boundary and interface conditions. The results are based on an algebraic DMP for suitable systems of ordinary differential equations. © 2012 The author 2012. Published by Oxford University...

In this paper we investigate the numerical solution of non-linear equations in an abstract (Banach space) setting. The main result is that the convergence can be guaranteed by two, directly checkable conditions (namely, by the consistency and the stability). We show that these conditions together are a sufficient, but not necessary condition for th...

Advection equations appear often in large-scale mathematical models arising in many fields of science and engineering. The Crank-Nicolson scheme can successfully be used in the numerical treatment of such equations. The accuracy of the numerical solution can sometimes be increased substantially by applying the Richardson Extrapolation. Two theorems...

For the Maxwell equations in time-dependent media only finite difference schemes with time-dependent conductivity are known. In this paper we present a numerical scheme based on the Magnus expansion and operator splitting that can handle time-dependent permeability and permittivity too. We demonstrate our results with numerical tests. KeywordsMaxw...

Initial value problems for systems of ordinary differential equations (ODEs) are solved numerically by using a combination of (a) the θ-method, (b) the sequential splitting procedure and (c) Richardson Extrapolation. Stability results for the combined numerical method are proved. It is shown, by using numerical experiments, that if the combined num...

The influence of future climatic changes on some high pollution levels that can cause damage to plants and human beings is studied in this paper. The particular area of interest is Hungary and its surrounding countries. Three important quantities, which are closely related to ozone concentrations, have been investigated. We shall mainly focus on ca...