Numerical solutions of kinetic equations on a spreadsheet

1987 ◽  
Vol 64 (6) ◽  
pp. 496 ◽  
Author(s):  
Douglas A. Coe
Keyword(s):  
Numerical Solutions ◽  
Kinetic Equations

A simple model for sustained oscillations in isothermal branched-chain or autocatalytic reactions in a well stirred open system - II. Limit cycles and non-stationary states

1985 ◽  
Vol 398 (1814) ◽  
pp. 101-116 ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Residence Time ◽  
Numerical Solutions ◽  
Kinetic Equations ◽  
Kinetic Scheme ◽  
Analytical Treatment ◽  
Chemical Systems ◽  
Sustained Oscillations ◽  
Gas Phase Oxidation ◽  
Branched Chain

The oscillatory patterns of behaviour exhibited by the simple kinetic scheme A + B → 2B; rate = k 1 ab , B → C; rate = k 2 b /(1 + rb ) are examined in detail. For systems with slowly decaying catalysts, such that k 2 ≪ k 1 a 2 0 a reduced asymptotic form of the governing equations allows a full analytical treatment. Oscillations begin as the residence time is increased through a point of Hopf bifurcation ז * res . The bifurcation is always supercritical, with the amplitudes of the concentration variations increasing from infinitesimally small values. The amplitudes grow initially as ( ז res – ז * res ) ½ , and tend to finite limiting magnitudes at very long residence times. At this limit, the oscillations in A have a ‘saw-tooth’ waveform, with B varying in a pulse-like manner. Numerical solutions of the full kinetic equations for a non-zero inflow of the catalyst B reveal how the system approaches a second Hopf bifurcation as the residence time is increased. The oscillatory amplitude now reaches a maximum and then decays back to zero. The applicability of this model to real chemical systems is discussed with particular reference to the gas phase oxidation of carbon monoxide.

Smoothed solutions in the kinetic theory of e+e- vacuum pair creation in strong laser fields. Linear polarization

2014 ◽  
Vol 23 (11) ◽  
pp. 1450068 ◽  
Author(s):  
S. A. Smolyansky ◽  
A. V. Prozorkevich ◽  
V. V. Dmitriev ◽  
A. V. Tarakanov
Keyword(s):  
Kinetic Theory ◽  
Time Dependence ◽  
Numerical Solutions ◽  
Kinetic Equations ◽  
Approximate Solutions ◽  
Plasma Phys ◽  
Intense Laser ◽  
Electron Positron ◽  
Important Open Problem ◽  
Focus Spot

In this paper, the dynamical Schwinger effect of vacuum creation of electron–positron pairs driven by an intense laser pulse is studied on the basis of correct quantum kinetic theory. In the general case, the numerical solutions of corresponding system of kinetic equations exhibit complex time dependence which makes the analysis of the physical processes complicated. In particular, the question of secondary effects, such as creation of annihilation photons from the focus spot of the colliding laser beams, remains an important open problem. In our previous work [S. A. Smolyansky, M. Bonitz and A. V. Prozorkevich, Contrib. Plasma Phys.53 (2013) 788], we presented a perturbation theory which is able to capture the dominant time dependence of the produced electron–positron pair distribution during the pulse (quasiparticle excitations). In the present work, we develop appreciably this approximation scheme. We demonstrate effectiveness of the proposed method for solution of such kind nonstationary problems in the simplest models of the laser field. However, this approach opens perspective for search of the relevant approximate solutions in kinetic theory of the e+e- quasiparticle plasma for the more realistic field models (arbitrary polarization, space inhomogeneous, etc).

KINETIC APPROACH TO THE INITIAL VALUE PROBLEM IN QUANTUM FIELD THEORY

1990 ◽  
Vol 05 (20) ◽  
pp. 1605-1614 ◽  
Author(s):  
LIN CHI YONG ◽  
A.F.R. DE TOLEDO PIZA
Keyword(s):  
Field Theory ◽  
Body Problem ◽  
Numerical Solutions ◽  
Kinetic Equations ◽  
Higher Dimensions ◽  
Quantum Field ◽  
Many Body ◽  
Initial Value ◽  
Projection Techniques ◽  
Working Out

Time-dependent projection techniques developed to derive kinetic equations in the context of the quantum many-body problem are applied to Φ4 field theory. The approach is illustrated by working out the (0+1)-dimensional case explicitly, including numerical solutions of the kinetic equations. Extension to higher dimensions is briefly discussed.

scholarly journals Algorithms for the computer simulation of ultrafast photoinduced intermolecular charge transfer in liquids

2020 ◽  
pp. 27-40
Author(s):  
С.В. Феськов ◽  
С.С. Хохлова
Keyword(s):  
Electron Transfer ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Kinetic Equations ◽  
Numerical Algorithms ◽  
Electronic States ◽  
Photochemical Reactions ◽  
Solution Of Equations ◽  
Donor Acceptor ◽  
Energy Surfaces

Предложены подходы к численному решению систем уравнений, описывающих кинетику двухстадийной фотохимической реакции в вязком полярном растворителе. Математическая модель построена на основе расширенной интегральной теории встреч и учитывает диффузионную подвижность молекул-реагентов в жидкости, неравновесность среды и внутримолекулярных степеней свободы, удаленный перенос электрона в донорно-акцепторных парах, разделенных растворителем. В рамках метода броуновского моделирования разработаны алгоритмы расчета безреакционных стохастических траекторий частиц на поверхностях свободной энергии, соответствующих различным состояниям реагентов и продуктов, схемы детектирования реакционных событий и генерации электронных прыжков, а также алгоритмы расчета нестационарных потоков частиц между электронными состояниями и вычисления интегральных ядер кинетических уравнений. Представлены результаты тестовых расчетов, демонстрирующие корректность численного решения и воспроизводящие известные особенности реакций электронного переноса в полярных жидкостях. Efficient approaches to the numerical solution of equations describing the kinetics of two-stage photochemical reactions in a viscous polar solvent are proposed. The mathematical model is based on the extended integral encounter theory and takes into account diffusive mobility of reactants in solution, nonequilibrium of solvent and intramolecular degrees of freedom, and remote electron transfer in solvent-separated donor-acceptor pairs. In the framework of the Brownian simulation technique, a number of numerical algorithms for calculating unreactive stochastic trajectories of particles on free energy surfaces corresponding to different electronic states of reactants and products are suggested, some computational schemes for the detection of reaction events and the generation of electronic hops are developed, and algorithms for calculating the time-dependent reaction fluxes between the electronic states and integral kernels of the kinetic equations are implemented. The results of test simulations demonstrating the validity of the numerical solutions and reproducing well-known features of electron transfer reactions in polar solvents are discussed.

Numerical Study of Partially Conservative Moment Equations in Kinetic Theory

2017 ◽  
Vol 21 (4) ◽  
pp. 981-1011 ◽  
Author(s):  
Julian Koellermeier ◽  
Manuel Torrilhon
Keyword(s):  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Numerical Study ◽  
Kinetic Equations ◽  
Conservative System ◽  
Moment Equations ◽  
Convergence Study ◽  
Moment Models ◽  
Model Equations ◽  
The Moment

AbstractMoment models are often used for the solution of kinetic equations such as the Boltzmann equation. Unfortunately, standard models like Grad's equations are not hyperbolic and can lead to nonphysical solutions. Newly derived moment models like the Hyperbolic Moment Equations and the Quadrature-Based Moment Equations yield globally hyperbolic equations but are given in partially conservative form that cannot be written as a conservative system.In this paper we investigate the applicability of different dedicated numerical schemes to solve the partially conservative model equations. Caused by the non-conservative type of equation we obtain differences in the numerical solutions, but due to the structure of the moment systems we show that these effects are very small for standard simulation cases. After successful identification of useful numerical settings we show a convergence study for a shock tube problem and compare the results to a discrete velocity solution. The results are in good agreement with the reference solution and we see convergence considering an increasing number of moments.

Gas-Kinetic Modeling and Simulation of Pedestrian Flows

2000 ◽  
Vol 1710 (1) ◽  
pp. 28-36 ◽  
Author(s):  
Serge Hoogendoorn ◽  
Piet H. L. Bovy
Keyword(s):  
Phase Space ◽  
Numerical Solutions ◽  
Kinetic Equations ◽  
Flow Simulation ◽  
Test Case ◽  
Phase Space Density ◽  
Pedestrian Flow ◽  
Space Density ◽  
Model Based ◽  
Vehicular Traffic Flow

Insight into pedestrian flow operations is important in both planning and geometric design of infrastructure facilities such as railway stations as well as in the management of pedestrian flows in such facilities. Lack of empirical knowledge regarding the characteristics of pedestrian flows under varying circumstances and designs motivates using a model-based approach. In this study, a new pedestrian flow model based on the gaskinetic modeling paradigm is established. The mesoscopic equations describe the dynamics of so-called pedestrian phase-space density, which can be considered as a two-dimensional generalization of the phase-space density used in gas-kinetic vehicular traffic flow. Convection, acceleration, and noncontinuum transition terms govern the dynamics. The latter terms reflect the dynamic influence of pedestrians decelerating and the changing angle of movement due to pedestrians interacting. Numerical solutions of the resulting gas-kinetic equations are established by using a novel particle discretization approach. Essentially, this approach upgrades the mesoscopic equations to a microscopic pedestrian flow simulation model. Using the particle discretization approach, the model’s behavior is tested for different test-case scenarios. The model is shown to produce plausible speed-density functions from which walking speeds and travel times can be derived for a variety of conditions.

Photodissolution of iron oxides in malonic acid

1994 ◽  
Vol 72 (10) ◽  
pp. 2037-2043 ◽  
Author(s):  
Marta I. Litter ◽  
Marina Villegas ◽  
Miguel A. Blesa
Keyword(s):  
Ascorbic Acid ◽  
Aqueous Solutions ◽  
Iron Oxides ◽  
Malonic Acid ◽  
Numerical Solutions ◽  
Kinetic Equations ◽  
Dark Reaction ◽  
Thermal Dissolution ◽  
Photochemical Initiation ◽  
Fitting Parameters

The influence of 254-nm irradiation on the dissolution rate of magnetite (Fe3O4) and maghemite (γ-Fe2O3) suspended in aqueous solutions of malonic acid is analyzed and compared with the dark reaction. The effect of the addition of Fe3+, reductants such as ascorbic acid and oxidants such as Ag+ or O2 is described. Photochemical initiation involves the production of >FeII by electrons photogenerated on the oxide, which triggers thermal dissolution. Experimental results are fitted by analytical and numerical solutions of the set of kinetic equations. The calculated fitting parameters reflect the lower activity of iron oxides in malonic acid compared to EDTA media.

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