Nonlinear monotonization of the Babenko scheme for the quasi‐linear advection equation

The paper is devoted to construction and development of new method for numerical solution of hyperbolic type equations [14, 17]. In the previous papers [4, 5, 6, 7, 8, 9] authors have investigated theoretically and tested experimentally 26 different finite‐difference schemes on 4 point patterns for the simplest hyperbolic equation: linear advection equation. This equation has the main features of every hyperbolic equation and is the important part of many mathematical models. In other cases the advection operator is the important part of the full operator of the problem. All 26 schemes have been compared experimentally on the special representative set of tests. Nevertheless to simplicity of the equation, almost all schemes have different disadvantages. They are discussed in detail in the cited papers. So, the investigation of new schemes for this equation is still an important task. In [4, 5, 6, 7, 8, 9] some new schemes were constructed for solving this advection equation. The nonlinear monotone Babenko scheme ("square") proved to be the best among all 26 schemes. So, it is a big interest to generalize this scheme to more difficult equations. The important example is a quasi‐linear advection equation. In this paper our basic aim is to construct a quasi‐monotone nonlinear Babenko scheme for solving the quasi‐linear advection equation and to test it experimentally. The monotonisation of the scheme is done by adding the artificial diffusion with limiters. We also present advanced results of comparative analysis of the new scheme with other known schemes. We have considered explicit and implicit upwind approximation schemes [4, 6, 13, 16] which is firstorder accurate in time and space, the Lax‐Wendroff scheme [4] which is the first order accurate in time and second order accurate in space. We also analyze the monotonised “Cabaret” scheme proposed in [10, 11]. It is second order accurate in time and space, and its monotonisation is based on apriori knowledge of the dependence region of the exact solution. The authors of this scheme called it by “jumping advection”. The considered schemes are compared numerically by using a set of tests, which is similar to one used in [4, 5, 6, 8]. Šiame straipsnyje pasiūlyta kvazi‐monotonie netiesine Babenkos skirtumu schema kvazitiesinei pernešimo lygčiai spresti. Schemos monotoniškumas pasiekiamas pridedant dirbtine difuzija su apribojimais. Pateiktas šios schemos palyginimas su kitomis schemomis. Taip pat analizuojama antros eiles pagal laika ir erdve monotonine “Cabaret” schema. Pateikti testu rezultatai. First Published Online: 14 Oct 201

Near infrared light emission quenching in organolanthanide complexes

We investigate the quenching of the near infrared light emission in Er3+ complexes induced by the resonant dipolar interaction between the rare-earth ion and high frequency vibrations of the organic ligand. The nonradiative decay rate of the lanthanide ion is discussed in terms of a continuous medium approximation, which depends only on a few, easily accessible spectroscopic and structural data. The model accounts well for the available experimental results in Er3+ complexes, and predicts an similar to 100% light emission quantum yield in fully halogenated systems

Natural Nuclear Reactor Oklo and Variation of Fundamental Constants Part 1: Computation of Neutronics of Fresh Core

Using modern methods of reactor physics we have performed full-scale calculations of the natural reactor Oklo. For reliability we have used recent version of two Monte Carlo codes: Russian code MCU REA and world wide known code MCNP (USA). Both codes produce similar results. We have constructed a computer model of the reactor Oklo zone RZ2 which takes into account all details of design and composition. The calculations were performed for three fresh cores with different uranium contents. Multiplication factors, reactivities and neutron fluxes were calculated. We have estimated also the temperature and void effects for the fresh core. As would be expected, we have found for the fresh core a significant difference between reactor and Maxwell spectra, which was used before for averaging cross sections in the Oklo reactor. The averaged cross section of Sm-149 and its dependence on the shift of resonance position (due to variation of fundamental constants) are significantly different from previous results. Contrary to results of some previous papers we find no evidence for the change of the fine structure constant in the past and obtain new, most accurate limits on its variation with time: -4 10^{-17}year^{-1} < d alpha/dt/alpha < 3 10^{-17} year^{-1} A further improvement in the accuracy of the limits can be achieved by taking account of the core burnup. These calculations are in progress.Comment: 25 pages, 14 figures, 12 tables, minor corrections, typos correcte

Spinoptics in a stationary spacetime

The main goal of the present paper is to study how polarization of photons affects their motion in a gravitational field created by a rotating massive compact object. We study propagation of the circularly polarized beams of light in a stationary gravitational field. We use (3+1)-form of the Maxwell equations to derive a master equation for the propagation of monochromatic electromagnetic waves of the frequency $\omega$ with a given helicity. We first analize its solutions in the high frequency approximation using the `standard' geometrical optics approach. After that we demonstrate how this `standard' approach can be modified in order to include the effect of the helicity of photons on their motion. Such an improved method reproduces the standard results of the geometrical optics at short distances. However, it modifies the asymptotic behavior of the circularly polarized beams in the late-time regime. We demonstrate that the corresponding equations for the circularly polarized beam can be effectively obtained by modification of the background geometry by including a small factor proportional to $\omega^{-1}$ whose sign corresponds to photon helicity. We obtain the modified equations for circularly polarized rays by using such a `renormalization' procedure, and calculate the corresponding renormalization term for the Kerr geometry.Comment: Added references, 13 page

Nonlinear monotonization of the Babenko scheme

The goal of the paper is to present and test the nonlinear monotonization of the Babenko scheme for solving 2D linear advection equation with alternating‐sign velocities. The numerical method of monotonization is based on the idea of limited artificial diffusion. There are some approaches for constructing quasi‐monotonic second order approximation schemes for solving hyperbolic systems and equations of gas dynamics: flux correction methods, the Godunov method, TVD methods and others. In particular, many authors developed the idea of TVD method. We try to use this idea to get a new quasi‐monotonic high order accuracy scheme based on the well‐known non‐monotonic Babenko scheme. The algorithm is presented for 1D problem. For testing 2D problem we use the splitting algorithm. The proposed monotonized scheme has shown the best results among all considered in the paper schemes especially for non‐smooth initial profile. Babenko schemos ("kvadrato") netiesinė monotonizacija Santrauka. Straipsnio tikslas yra Babenko schemos dvimačiam tiesiniam advekcijos uždaviniui su ženklą keičiančiais greičiais netiesinės monotonizacijos metodo pateikimas ir testavimas. Skaitinis monotonizacijos metodas remiasi dirbtinės difuzijos įvedimo idėja. Egzistuoja keli kvazimonotoniniu antros aproksimacijos eilės schemų hiperbolinems sistemoms ir dujų dinamikos lygtims konstravimo būdai: srautų korekcijos metodas, Godunovo metodas, TVD ir kiti metodai. Mes naudojame TVD idėją naujos kvazimonotoninės aukštos tikslumo eilės schemos gavimui remiantis plačiai žinoma monotonine baigtinių skirtumų Babenko schema. Skaitinis algoritmas pateiktas vienmačio uždavinio atveju. Dvimačio uždavinio sprendimui taikomas faktorizacijos algoritmas. Pasiūlytos monotonizuotos schemos pagalba gauti rezultatai yra geriausi, lyginant su kitu straipsnyje naudojamų schemų skaičiavimų rezultatais. Ypatingai gerai tai matoma neglodaus pradinio profilio atveju. First Published Online: 14 Oct 201