On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

Schwarz waveform relaxation algorithms are designed for the linear Schrödinger equation with potential. Two classes of algorithms are introduced: the quasi-optimal algorithm, based on the transparent continuous or discrete boundary condition, and the optimized complex Robin algorithm. We analyze their properties in one dimension. First, well-posedness and convergence are studied, in the overlapping and the non-overlapping case, for constant or non-constant potentials. Then discrete algorithms are established, for which convergence is proved through discrete energies or Fourier transforms, as in the continuous case. Numerical results illustrate the efficiency of the methods, for various types of potentials and any number of subdomains.

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Schwarz waveform relaxation method for one-dimensional Schrödinger equation with general potential

Numerical Algorithms ◽

10.1007/s11075-016-0153-4 ◽

2016 ◽

Vol 74 (2) ◽

pp. 393-426 ◽

Cited By ~ 7

Author(s):

Christophe Besse ◽

Feng Xing

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Relaxation Method ◽

Waveform Relaxation ◽

One Dimensional ◽

Schwarz Waveform Relaxation ◽

Waveform Relaxation Method ◽

General Potential ◽

Schwarz Waveform Relaxation Method

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Optimized and Quasi-optimal Schwarz Waveform Relaxation for the One Dimensional Schrödinger Equation

Lecture Notes in Computational Science and Engineering - Domain Decomposition Methods in Science and Engineering XVII ◽

10.1007/978-3-540-75199-1_24 ◽

2008 ◽

pp. 221-228 ◽

Cited By ~ 2

Author(s):

Laurence Halpern ◽

Jérémie Szeftel

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Waveform Relaxation ◽

One Dimensional ◽

Schwarz Waveform Relaxation ◽

The One

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Introduction

10.1093/oso/9780198805014.003.0001 ◽

2018 ◽

Author(s):

Niels Engholm Henriksen ◽

Flemming Yssing Hansen

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Degrees Of Freedom ◽

State Level ◽

Boltzmann Distribution ◽

Theoretical Description ◽

Time Dependent ◽

Nuclear Dynamics ◽

Molecular Reaction ◽

Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

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