A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

AbstractMany simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge–Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic orde

Numerov and phase-integral methods for charmonium

This paper applies the Numerov and phase-integral methods to the stationary Schrödinger equation that studies bound states of charm anti-charm quarks. The former is a numerical method well suited for a matrix form of the second-order ordinary differential equations, and can be applied whenever the stationary states admit a Taylor-series expansion. The latter is an analytic method that provides, in principle, even exact solutions of the stationary Schrödinger equation, and well suited for applying matched asymptotic expansions and higher-order quantization conditions. The Numerov method is found to be always in agreement with the early results of Eichten et al., whereas an original evaluation of the phase-integral quantization condition clarifies under which conditions the previous results in the literature on higher-order terms can be obtained

On supraconvergence phenomenon for second order centered finite differences on non-uniform grids

In the present study we consider an example of a boundary value problem for a simple second order ordinary differential equation, which may exhibit a boundary layer phenomenon. We show that usual central finite differences, which are second order accurate on a uniform grid, can be substantially upgraded to the fourth order by a suitable choice of the underlying non-uniform grid. This example is quite pedagogical and may give some ideas for more complex problems.Comment: 26 pages, 2 figures, 2 tables, 37 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

A New Trigonometrically Fitted Two-Derivative Runge-Kutta Method for the Numerical Solution of the Schrödinger Equation and Related Problems

A new trigonometrically fitted fifth-order two-derivative Runge-Kutta method with variable nodes is developed for the numerical solution of the radial Schrödinger equation and related oscillatory problems. Linear stability and phase properties of the new method are examined. Numerical results are reported to show the robustness and competence of the new method compared with some highly efficient methods in the recent literature

De la phénoménologie à la microscopie, une nouvelle approche pour l’évaluation des sections efficaces de fission

The work presented here aims to improve models used in the fission crosssectionevaluation. The results give insights for a significant breakthrough in this fieldand yielded large extensions of the evaluation code CONRAD. Partial cross sections areinherently strongly correlated together as of the competition of the related reactions mustyield the total cross section. Therefore improving fission cross section benefits to all partialcross sections. A sound framework for the simulation of competitive reactions hadto be settled in order to further investigate on the fission reaction; this was implementedusing the TALYS reference code as guideline. After ensuring consistency and consistencyof the framework, focus was made on fission. Perspective resulting from the useof macroscopic-microscopic models such as the FRDM and FRLDM were analyzed; thesemodels have been implemented and validated on experimental data and benchmarks. Tocomply with evaluation requirements in terms of computation time, several specific numericalmethods have been used and parts of the program were written to run on GPU.These macroscopic-microscopic models yield potential energy surfaces that can be used toextract a one-dimensional fission barrier. This latter can then be used to obtained fissiontransmission coefficients that can be used in a Hauser-Feshbach model. This method hasbeen finally tested for the calculation of the average fission cross section for 239Pu(n,f).Les travaux présentés visent à améliorer les modèles de physique nucléaireutilisés dans l’évaluation des sections efficaces neutroniques de fission. Le résultat deces travaux donne les clefs pour une percée significative dans ce domaine et a permisd’étendre fortement les capacités du code d’évaluation CONRAD. Les sections partiellesétant naturellement corrélées entre-elles pour respecter la valeur de la section totale, cesaméliorations bénéficient à l’ensemble des sections partielles. Un cadre solide pour lamodélisation des processus concurrent à la fission a dû être établi sur le modèle du codede référence TALYS. Après s’être assuré de la fiabilité et de la cohérence du cadre, lesinvestigations spécifiques concernant la fission ont pu être réalisées. Les perspectivesd’applications offertes par les modèles macro-microscopiques FRDM et FRLDM ont étéanalysées. Ces modèles ont été implémentés et validés sur des données expérimentaleset des benchmarks. Afin d’obtenir des temps de calcul compatibles avec les besoins del’évaluation, des méthodes numériques sophistiquées ont été sélectionnées et une partiedes calculs a été portée sur GPU. Ces modèles macro-microscopiques peuvent être utiliséspour construire des surfaces d’énergie potentielle qui sont à leur tour traitées afin d’obtenirdes barrières de fission à une dimension, puis des coefficients de transmission fission. Cesderniers sont alors utilisés dans le cadre de modélisation des sections efficaces moyennesdu domaine statistique sur la base d’un modèle Hauser-Feshbach. Les résultats de cetteapproche seront présentés sur le cas du 239Pu(n,f)

Mathematical Modeling with Differential Equations in Physics, Chemistry, Biology, and Economics

This volume was conceived as a Special Issue of the MDPI journal Mathematics to illustrate and show relevant applications of differential equations in different fields, coherently with the latest trends in applied mathematics research. All the articles that were submitted for publication are valuable, interesting, and original. The readers will certainly appreciate the heterogeneity of the 10 papers included in this book and will discover how helpful all the kinds of differential equations are in a wide range of disciplines. We are confident that this book will be inspirational for young scholars as well