Exponentially fitted open Newton–Cotes differential methods as multilayer symplectic integrators

2010 ◽  
Vol 132 (20) ◽  
pp. 204107 ◽  
Author(s):  
G. Vanden Berghe ◽  
M. Van Daele
Keyword(s):  
Symplectic Integrators ◽  
Exponentially Fitted ◽  
Differential Methods

Closed Newton-Cotes Trigonometrically-Fitted Formulae for Numerical Integration of the Schrödinger Equation

2007 ◽  
Vol 3 (1) ◽  
pp. 45-57 ◽  
Author(s):  
T.E. Simos
Keyword(s):  
Numerical Integration ◽  
Symplectic Geometry ◽  
Efficient Solution ◽  
Symplectic Integrators ◽  
Multilayer Structures ◽  
Symplectic Methods ◽  
One Dimensional ◽  
One Step ◽  
Differential Methods ◽  
Symplectic Schemes

In this paper we investigate the connection between closed Newton-Cotes formulae, trigonometrically-fitted differential methods, symplectic integrators and efficient solution of the Schr¨odinger equation. Several one step symplectic integrators have been produced based on symplectic geometry, as one can see from the literature. However, the study of multistep symplectic integrators is very poor. Zhu et. al. [1] has studied the symplectic integrators and the well known open Newton-Cotes differential methods and as a result has presented the open Newton-Cotes differential methods as multilayer symplectic integrators. The construction of multistep symplectic integrators based on the open Newton-Cotes integration methods was investigated by Chiou and Wu [2]. In this paper we investigate the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. We also develop trigonometrically-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the symplectic schemes to the well known one-dimensional Schr¨odinger equation in order to investigate the efficiency of the proposed method to these type of problems.

CLOSED NEWTON–COTES TRIGONOMETRICALLY-FITTED FORMULAE FOR LONG-TIME INTEGRATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1061-1074 ◽  
Author(s):  
T. E. SIMOS
Keyword(s):  
Symplectic Geometry ◽  
Equations Of Motion ◽  
Time Integration ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Long Time ◽  
One Step ◽  
Long Time Integration ◽  
Differential Methods ◽  
Symplectic Schemes

The connection between closed Newton–Cotes, trigonometrically-fitted differential methods and symplectic integrators is investigated in this paper. It is known from the literature that several one-step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. Zhu et al.2 presented the well known open Newton–Cotes differential methods as multilayer symplectic integrators. Chiou and Wu2 also investigated the construction of multistep symplectic integrators based on the open Newton–Cotes integration methods. In this paper we investigate the closed Newton–Cotes formulae and we write them as symplectic multilayer structures. After this we construct trigonometrically-fitted symplectic methods which are based on the closed Newton–Cotes formulae. We apply the symplectic schemes in order to solve Hamilton's equations of motion which are linear in position and momentum. We observe that the Hamiltonian energy of the system remains almost constant as integration procceeds.

Assessing the competitiveness of the resort area

2020 ◽  
Vol 18 (7) ◽  
pp. 1216-1231
Author(s):  
K.A. Nefedova ◽  
D.O. Maslakova
Keyword(s):  
Comparative Analysis ◽  
Competitive Ability ◽  
Effective Management ◽  
Development Indicators ◽  
Ski Resort ◽  
Ski Resorts ◽  
Ranking Score ◽  
Comprehensive Development ◽  
Differential Methods

Subject. This article discusses the issues of development of the Krasnaya Polyana resort area. Objectives. The article aims to assess the competitive ability and attractiveness of this resort area through developing indicators. Methods. For the study, we used a comparative analysis, and factor and ranking score techniques. Results. The article offers original methods to assess the competitiveness of the ski resort area and describes possible directions to increase and improve the competitiveness and attractiveness of ski resorts. Conclusions. Comprehensive development indicators help assess the competitive ability of the ski resort area. Modified expert, sociological, rating, and differential methods contribute to the effective management of the resort area's advantages.

Hamilton-Jacobi Theory

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Bbgky Hierarchy ◽  
Distribution Functions ◽  
Symplectic Integrators ◽  
Partition Functions ◽  
Von Neumann ◽  
Equipartition Theorem ◽  
Recurrence Theorem ◽  
Phase Amplitude ◽  
Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

Symplectic integrators with adaptive time step applied to runaway electron dynamics

2019 ◽  
Vol 81 (4) ◽  
pp. 1295-1309 ◽  
Author(s):  
Yanyan Shi ◽  
Yajuan Sun ◽  
Yang He ◽  
Hong Qin ◽  
Jian Liu
Keyword(s):  
Runaway Electron ◽  
Symplectic Integrators ◽  
Electron Dynamics ◽  
Time Step ◽  
Adaptive Time Step ◽  
Adaptive Time
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