Using preconditioned adaptive step size Runge-Kutta methods for solving the time-dependent Schrödinger equation

2004 ◽  
Vol 121 (23) ◽  
pp. 11535-11541 ◽  
Author(s):  
Jean Christophe Tremblay ◽  
Tucker Carrington
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
Step Size ◽  
Adaptive Step Size ◽  
Runge Kutta ◽  
Adaptive Step

scholarly journals Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 374
Author(s):  
Athinoula A. Kosti ◽  
Simon Colreavy-Donnelly ◽  
Fabio Caraffini ◽  
Zacharias A. Anastassi
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Time Dependent ◽  
Computational Techniques ◽  
Efficient Computation ◽  
Maximum Absolute Error ◽  
Step Size ◽  
Nonlinear Schrödinger

Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations.

Introduction

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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