Метод конечных разностей во временной области для кусочно-однородных диэлектрических сред

In this paper, we consider a numerical solution of Maxwell’s curl equations for piecewise uniform dielectric medium by the example of a one-dimensional problem. For obtaining the second order accuracy, the electric field grid node is placed into the permittivity discontinuity point of the medium. If the dielectric permittivity is large, the problem becomes singularly perturbed and a contrast structure appears. We propose a piecewise quasi-uniform mesh which resolves all characteristic solution parts of the problem (regular part, boundary layer and transition zone placed between them) in detail. The features of the mesh are discussed. В данной статье рассматривается численное решение системы вихревых уравнений Максвелла для кусочно-однородной диэлектрической среды на примере одномерной задачи. Для обеспечения второго порядка точности необходимо поставить узел сетки электрического поля в точку разрыва диэлектрической проницаемости. Если скачок проницаемости велик, то задача становится сингулярно возмущенной и возникает контрастная структура. Построена кусочная квазиравномерная сетка, детально передающая все характерные участки решения этой задачи (регулярную область, пограничный слой и переходную зону между ними). Обсуждаются свойства этой сетки

Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point

Turning points occur in many circumstances in fluid mechanics. When the viscosity is small, very complex phenomena can occur near turning points, which are not yet well understood. A model problem, corresponding to a linear convection-diffusion equation (e.g., suitable linearization of the Navier-Stokes or B́nard convection equations) is considered. Our analysis shows the diversity and complexity of behaviors and boundary or interior layers which already appear for our equations simpler than the Navier-Stokes or B́nard convection equations. Of course the diversity and complexity of these structures will have to be taken into consideration for the study of the nonlinear problems. In our case, at this stage, the full theoretical (asymptotic) analysis is provided. This study is totally new to the best of our knowledge. Numerical treatment and more complex problems will be considered elsewhere.open91