scholarly journals High Order Split Operators for the Time-Dependent Wavepacket method of Triatomic Reactive Scattering in Hyperspherical Coordinates

Entropy ◽  
2019 ◽  
Vol 21 (10) ◽  
pp. 979
Author(s):  
Umair Umer ◽  
Hailin Zhao ◽  
Syed Kazim Usman ◽  
Zhigang Sun
Keyword(s):  
High Order ◽  
Time Dependent ◽  
Reactive Scattering ◽  
Hyperspherical Coordinates ◽  
Total Reaction ◽  
Sine Transform ◽  
Split Operator ◽  
Intermediate States ◽  
The Fourier Transform ◽  
Derivatives Of

Since the introduction of a series of methods for solving the time-dependent Schrödinger equation (TDSE) in the 80s of the last centry, such as the Fourier transform, the split operator (SO), the Chebyshev polynomial propagator, and complex absorbing potential, investigation of the molecular dynamics within quantum mechanics principle have become popular. In this paper, the application of the time-dependent wave packet (TDWP) method using high-order SO propagators in hyperspherical coordinates for solving triatomic reactive scattering was investigated. The fast sine transform was applied to calculate the derivatives of the wave function of the radial degree of freedom. These high-order SO propagators are examined in different forms, i.e., TVT (Kinetic–Potential–Kinetic) and VTV (Potential–Kinetic–Potential) forms with three typical triatomic reactions, H + H 2 , O + O 2 and F + HD. A little difference has been observed among the performances of high-order SO propagators in the TVT and VTV representations in the hyperspherical coordinate. For obtaining total reaction probabilities with 1% error, some of the S class high-order SO propagators, which have symmetric forms, are more efficient than second order SO for reactions involving long lived intermediate states. High order SO propagators are very efficient for obtaining total reaction probabilities.

Theoretical investigations of the Au++H2 reactive scattering by the time-dependent quantum wave packet method

2017 ◽  
Vol 31 (06) ◽  
pp. 1750039 ◽  
Author(s):  
Wentao Lee ◽  
Haixiang He ◽  
Maodu Chen
Keyword(s):  
Wave Packet ◽  
Cross Sections ◽  
Collision Energy ◽  
Time Dependent ◽  
Reactive Scattering ◽  
Total Reaction ◽  
Theoretical Investigations ◽  
Classical Trajectories ◽  
Quantum Wave

Employing the state-to-state time-dependent quantum wave packet method, the Au[Formula: see text]H2 reactive scattering with initial states [Formula: see text], [Formula: see text] and 1 were investigated. Total reaction probabilities, product state-resolved integral cross-sections (ICSs) and differential cross-sections (DCSs) were calculated up to collision energy of 4.5 eV. The numerical results show that total reaction probabilities and ICSs increase with increasing collision energies, and there is little effect to the reactive scattering processes from the rotational excitation of H2 molecule. Below collision energy of around 3.0 eV, the role of the potential well in the entrance channel is significant and the reactive scattering proceeds dominantly by an indirect process, which leads to a nearly symmetric shape of the DCSs. With collision energy higher than 4.0 eV, the reactive scattering proceeds through a direct process, which leads to a forward biased DCSs, and also a hotter rotational distributions of the products. Total ICS agrees with the results by the quasi-classical trajectories theory very well, which suggests that the quantum effects in this reactive process are not obvious. However, the agreement between the experimental total cross-section and our theoretical result is not so good. This may be due to the uncertainty of the experiment or/and the inaccuracy of the potential energy surface.

scholarly journals Extended time-dependent mild-slope and wave-action equations for wave–bottom and wave–current interactions

2011 ◽  
Vol 468 (2137) ◽  
pp. 184-205 ◽  
Author(s):  
Yaron Toledo ◽  
Tai-Wen Hsu ◽  
Aron Roland
Keyword(s):  
Type Equation ◽  
Wave Action ◽  
High Order ◽  
Time Dependent ◽  
Mild Slope Equation ◽  
Wave Ray ◽  
Ambient Currents ◽  
Order Components ◽  
Wave Forecasting ◽  
Derivatives Of

Extended mild-slope (MS) and wave-action equations (WAEs) are derived by taking into account high-order derivatives of the bottom profile and the depth-averaged current that were previously neglected. As a first step for this derivation, a time-dependent MS-type equation in the presence of ambient currents that consists of these high-order components is constructed. This mild-slope equation is used as a basis to form a wave-action balance equation that retains high-order refraction and diffraction terms of varying depths and currents. The derivation accurately accounts for the effects of the currents on the Doppler shift. This results in an ‘effective’ intrinsic frequency and wavenumber that differ from the ones of wave ray theory. Finally, the new WAE is derived for the phase-averaged frequency-direction spectrum in order to allow its use in stochastic wave-forecasting models.

A Note on Fourier Transforms and Imbedding Theorems

1967 ◽  
Vol 10 (5) ◽  
pp. 695-698
Author(s):  
Robert A. Adams
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
High Order ◽  
Simple Consequence ◽  
The Fourier Transform ◽  
Derivatives Of

It is well known that Sobolev′s Lemma on the continuity of functions possessing L2 distributional derivatives of sufficiently high order is a simple consequence of elementary properties of the Fourier transform in L2 (e.g. [1, p. 174]). (In fact this statement remains true if 2 is replaced by p, 1 ≤ p ≤ 2). In this note we show that imbedding theorems of the type Wm, p ⊂Lq can also be obtained using Fourier transforms and an elementary lemma which reduces the cases p > 2 to the case p = 2. The simplicity of this approach is obtained at the expense of a slight loss of generality in the imbedding theorem.

scholarly journals Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1113
Author(s):  
Isaías Alonso-Mallo ◽  
Ana M. Portillo
Keyword(s):  
Numerical Experiments ◽  
Splitting Method ◽  
Method Of Lines ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
High Order ◽  
Time Dependent ◽  
Boundary Values ◽  
Lipschitz Continuous

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

scholarly journals High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

2021 ◽  
Author(s):  
Giacomo Albi ◽  
Lorenzo Pareschi
Keyword(s):  
Multistep Methods ◽  
General Setting ◽  
Reaction Diffusion ◽  
Strong Stability ◽  
High Order ◽  
Iterative Solvers ◽  
Time Dependent ◽  
Linear Multistep Methods ◽  
Advection Diffusion Equation ◽  
Separation Of Scales

AbstractWe consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

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