scholarly journals A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1109
Author(s):  
N. Mohankumar ◽  
Tucker Carrington
Keyword(s):  
Error Bound ◽  
A Priori ◽  
Schroedinger Equation ◽  
Time Dependent ◽  
Comparison Of Methods ◽  
Lanczos Algorithm ◽  
Imaginary Time ◽  
Time Step ◽  
Time Propagation ◽  
Numer Anal

To use the short iterative Lanczos algorithm to solve the time-dependent Schroedinger equation, one must choose, for a given Lanczos space size, a time step. We compare the derivation of the well-known Lubich and Hochbruck time step from SIAM J. Numer. Anal. 34 (1997) 1911 with the a priori time step we proposed in Mohankumar and Carrington (MC) Comput. Phys. Commun., 181 (2010) 1859 and demonstrate that the MC time step is somewhat larger, i.e., that the MC error bound is tighter. In addition, we use the MC approach to derive an error bound and time step for imaginary time propagation. The error bound we derive is much tighter than the error bound of Stewart and Leyk.

EXPONENTIAL PROPAGATORS (INTEGRATORS) FOR THE TIME-DEPENDENT SCHRÖDINGER EQUATION

2013 ◽  
Vol 12 (06) ◽  
pp. 1340001 ◽  
Author(s):  
ANDRÉ D. BANDRAUK ◽  
HUIZHONG LU
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Derivative ◽  
Quantum Statistical Mechanics ◽  
Time Dependent ◽  
High Order Accuracy ◽  
Parabolic Partial Differential Equation ◽  
Imaginary Time ◽  
Time Step ◽  
Order Accuracy

The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.

scholarly journals ANALYTIC–NUMERICAL SOLUTIONS WITH A PRIORI BOUNDS FOR MATRIX–VECTOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

2002 ◽  
Vol 45 (1) ◽  
pp. 5-25 ◽  
Author(s):  
Lucas Jódar ◽  
Marco Marletta
Keyword(s):  
Numerical Solutions ◽  
A Priori ◽  
Approximate Solutions ◽  
Time Dependent ◽  
Vector System ◽  
Approximation Procedure ◽  
Time Step ◽  
Dependent Case ◽  
Independent Case ◽  
Matrix Vector

AbstractWe consider a parabolic matrix–vector system in which the diffusion matrix may be time dependent. For the time-independent case we construct approximate solutions with guaranteed error bounds using spectral information from certain matrix–vector Sturm–Liouville problems. For the time-dependent case we employ an approximation procedure which reduces the problem, on each time-step, to the time-independent case. We give an algorithm which may be used a priori at each time-step in the time-dependent case to guarantee accuracy to a specified tolerance.AMS 2000 Mathematics subject classification: Primary 35C10; 35M10. Secondary 15A

SOLVING THE SINGLE LAYER MULTICONFIGURATION TIME-DEPENDENT HARTREE–FOCK EQUATION IN SECOND QUANTIZATION REPRESENTATION BASED ON A DECOMPOSITION OF THE OVERALL FOCK SPACE

2013 ◽  
Vol 12 (01) ◽  
pp. 1250105 ◽  
Author(s):  
WENLIANG LI ◽  
X. WENWU ◽  
H. KELI
Keyword(s):  
Fock Space ◽  
Single Layer ◽  
Time Dependent ◽  
Initial Condition ◽  
Imaginary Time ◽  
Second Quantization ◽  
Hartree Fock ◽  
Time Propagation ◽  
Set Up ◽  
Quantization Representation

Very recently, the multilayer multiconfiguration time-dependent Hartree (MCTDH) method based on a decomposition of the overall Fock space in second quantization representation was proposed [Wang H, Thoss M, J Chem Phys131:024114, 2009], and they have presented demonstrative numerical example on vibrationally coupled electron transport. Followed by the thinking, the strategies of the implement of the theory to deal with multielectron dynamics with consideration coulomb interaction were discussed in detail. Some demonstrative calculations have been carried out by using the single layer MCTDHF method. The influences of the different choice way of initial condition, such as the species of basis function, the type of decomposition of the full Fock space, the choice of the spin orbitals, the number of the single particle function and so on, on the imaginary time propagation are mainly studied. The bridge between the generalized theory and the calculation application has been set up.

scholarly journals Statistical Uncertainty of DNS in Geometries without Homogeneous Directions

2021 ◽  
Vol 11 (4) ◽  
pp. 1399
Author(s):  
Jure Oder ◽  
Cédric Flageul ◽  
Iztok Tiselj
Keyword(s):  
Channel Flow ◽  
A Priori ◽  
Time Integration ◽  
Navier Stokes ◽  
Statistical Uncertainty ◽  
Time Step ◽  
Order Of Magnitude ◽  
Averaging Time ◽  
The Individual ◽  
Time Required

In this paper, we present uncertainties of statistical quantities of direct numerical simulations (DNS) with small numerical errors. The uncertainties are analysed for channel flow and a flow separation case in a confined backward facing step (BFS) geometry. The infinite channel flow case has two homogeneous directions and this is usually exploited to speed-up the convergence of the results. As we show, such a procedure reduces statistical uncertainties of the results by up to an order of magnitude. This effect is strongest in the near wall regions. In the case of flow over a confined BFS, there are no such directions and thus very long integration times are required. The individual statistical quantities converge with the square root of time integration so, in order to improve the uncertainty by a factor of two, the simulation has to be prolonged by a factor of four. We provide an estimator that can be used to evaluate a priori the DNS relative statistical uncertainties from results obtained with a Reynolds Averaged Navier Stokes simulation. In the DNS, the estimator can be used to predict the averaging time and with it the simulation time required to achieve a certain relative statistical uncertainty of results. For accurate evaluation of averages and their uncertainties, it is not required to use every time step of the DNS. We observe that statistical uncertainty of the results is uninfluenced by reducing the number of samples to the point where the period between two consecutive samples measured in Courant–Friedrichss–Levy (CFL) condition units is below one. Nevertheless, crossing this limit, the estimates of uncertainties start to exhibit significant growth.

Convergence for Imaginary Time Step evolution in the Fermi and Dirac seas

2010 ◽  
Vol 53 (2) ◽  
pp. 327-330 ◽  
Author(s):  
FangQiong Li ◽  
Ying Zhang ◽  
Jie Meng
Keyword(s):  
Imaginary Time ◽  
Time Step

scholarly journals Dynamical analysis of a reduced model for the North Atlantic Oscillation

2021 ◽  
Author(s):  
Courtney Quinn ◽  
Dylan Harries ◽  
Terence J. O’Kane
Keyword(s):  
North Atlantic Oscillation ◽  
North Atlantic ◽  
Growth Rates ◽  
Time Dependent ◽  
Unstable State ◽  
Time Step ◽  
Atlantic Sector ◽  
The North ◽  
The North Atlantic ◽  
The North Atlantic Oscillation

AbstractThe dynamics of the North Atlantic Oscillation (NAO) are analyzed through a data-driven model obtained from atmospheric reanalysis data. We apply a regularized vector autoregressive clustering technique to identify recurrent and persistent states of atmospheric circulation patterns in the North Atlantic sector (110°W-0°E, 20°N-90°N). In order to analyze the dynamics associated with the resulting cluster-based models, we define a time-dependent linear delayed map with a switching sequence set a priori by the cluster affiliations at each time step. Using a method for computing the covariant Lyapunov vectors (CLVs) over various time windows, we produce sets of mixed singular vectors (for short windows) and approximate the asymptotic CLVs (for longer windows). The growth rates and alignment of the resulting time-dependent vectors are then analyzed. We find that the window chosen to compute the vectors acts as a filter on the dynamics. For short windows, the alignment and changes in growth rates are indicative of individual transitions between persistent states. For long windows, we observe an emergent annual signal manifest in the alignment of the CLVs characteristic of the observed seasonality in the NAO index. Analysis of the average finite-time dimension reveals the NAO− as the most unstable state relative to the NAO+, with persistent AR states largely stable. Our results agree with other recent theoretical and empirical studies that have shown blocking events to have less predictability than periods of enhanced zonal flow.

Imaginary-Time Time-Dependent Density Functional Calculation of Excited States of Atoms Using CWDVR Approach

2021 ◽  
Vol 323 ◽  
pp. 14-20
Author(s):  
Naranchimeg Dagviikhorol ◽  
Munkhsaikhan Gonchigsuren ◽  
Lochin Khenmedekh ◽  
Namsrai Tsogbadrakh ◽  
Ochir Sukh
Keyword(s):  
Excited States ◽  
Density Functional ◽  
Time Dependent ◽  
Discrete Variable ◽  
Coulomb Wave Function ◽  
Discrete Variable Representation ◽  
Density Functional Calculation ◽  
Imaginary Time ◽  
Sham Equation ◽  
Variable Representation

We have calculated the energies of excited states for the He, Li, and Be atoms by the time dependent self-consistent Kohn Sham equation using the Coulomb Wave Function Discrete Variable Representation CWDVR) approach. The CWDVR approach was used the uniform and optimal spatial grid discretization to the solution of the Kohn-Sham equation for the excited states of atoms. Our results suggest that the CWDVR approach is an efficient and precise solutions of excited-state energies of atoms. We have shown that the calculated electronic energies of excited states for the He, Li, and Be atoms agree with the other researcher values.

scholarly journals Doubly Degenerate Parabolic Equation with Time-Dependent Gradient Source and Initial Data Measures

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Lihua Deng ◽  
Xianguang Shang
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Degenerate Parabolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Existence ◽  
Time Dependent ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

This paper is devoted to the Cauchy problem for a class of doubly degenerate parabolic equation with time-dependent gradient source, where the initial data are Radon measures. Using the delicate a priori estimates, we first establish two local existence results. Furthermore, we show that the existence of solutions is optimal in the class considered here.

Sign in / Sign up

Export Citation Format

Share Document