Mathematical treatment of environmental models

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 par- tial differential equations, which are defined on very large spatial domains and have to be solved numerically on very long time intervals. Moreover, very often many different scenarios have also to be developed and used in the investigations. There- fore, both the storage requirements and the computational work are enormous. The great difficulties can be overcome only if the following four tasks are successfully resolved: (a) fast and sufficiently accurate numerical methods are to be selected, (b) reliable and efficient splitting procedures are to be applied, (c) the cache memories of the available computers are to be efficiently exploited and (d) the codes are to be parallelized

HIGH ACCURACY FREQUENCY DETERMINATION FROM DISCRETE SPECTRA

The problem of determining the characteristics of a sine wave from its discrete spectrum is considered. The nontriviality of the problem is caused basicly by a phenomenon called spectral leakage, that is, by the fact that the spectral envelope of a single sinusoid forms a bell-shaped curve, even in the ideal noiseless case. In the paper a simple and self-contained treatment of spectral leakage is presented and a computationally efficient frequency estimation method is derived, taking into consideration different types of time-domain windows

Discrete maximum principles for nonlinear parabolic PDE systems

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

An IMEX scheme combined with Richardson extrapolation methods for some reaction-diffusion equations

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

On modifications of continuous and discrete maximum principles for reaction-diffusion problems

In this work, we present and discuss some modifications, in the form of two-sided estimation (and also for arbitrary source functions instead of usual sign-conditions), of continuous and discrete maximum principles for the reactiondiffusion problems solved by the finite element and finite difference methods

On continuous and discrete maximum principles for elliptic problems with the third boundary condition

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