scholarly journals Solution of two-electron Schrödinger equations using a residual minimization method and one-dimensional basis functions

AIP Advances ◽  
2021 ◽  
Vol 11 (2) ◽  
pp. 025228
Author(s):  
Faiz Ur Rahman ◽  
Yanoar Pribadi Sarwono ◽  
Rui-Qin Zhang
Keyword(s):  
Basis Functions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Minimization Method ◽  
One Dimensional ◽  
Residual Minimization

scholarly journals REDUCED ORDER MODELS BASED ON POD METHOD FOR SCHRÖDINGER EQUATIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 694-707 ◽  
Author(s):  
Gerda Jankevičiutė ◽  
Teresė Leonavičienė ◽  
Raimondas Čiegis ◽  
Andrej Bugajev
Keyword(s):  
Numerical Experiments ◽  
Approximate Solutions ◽  
Basis Functions ◽  
Schrodinger Equations ◽  
Reduced Order Models ◽  
Schrödinger Equations ◽  
Optimal Basis ◽  
Reduced Order ◽  
One Dimensional ◽  
Proper Orthogonal

Reduced-order models (ROM) are developed using the proper orthogonal decomposition (POD) for one dimensional linear and nonlinear Schrödinger equations. The main aim of this paper is to study the accuracy and robustness of the ROM approximations. The sensitivity of generated optimal basis functions on various parameters of the algorithms is discussed. Errors between POD approximate solutions and exact problem solutions are calculated. Results of numerical experiments are presented.

LOD-MS for Gross-Pitaevskii Equation in Bose-Einstein Condensates

2013 ◽  
Vol 14 (1) ◽  
pp. 219-241 ◽  
Author(s):  
Linghua Kong ◽  
Jialin Hong ◽  
Jingjing Zhang
Keyword(s):  
Energy Conservation ◽  
Computational Cost ◽  
Three Dimensional ◽  
Schrodinger Equations ◽  
Pitaevskii Equation ◽  
Schrödinger Equations ◽  
One Dimensional ◽  
Exactly Solvable ◽  
Bose Einstein ◽  
Energy Conservation Law

AbstractThe local one-dimensional multisymplectic scheme (LOD-MS) is developed for the three-dimensional (3D) Gross-Pitaevskii (GP) equation in Bose-Einstein condensates. The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional (LOD) method. The 3D GP equation is split into three linear LOD Schrödinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrödinger equations are multisymplectic which can be approximated by multisymplectic integrator (MI). The conservative properties of the proposed scheme are investigated. It is mass-preserving. Surprisingly, the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable. This is impossible for conventional MIs in nonlinear Hamiltonian context. The numerical results show that the LOD-MS can simulate the original problems very well. They are consistent with the numerical analysis.

scholarly journals Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities

2011 ◽  
Vol 10 (5) ◽  
pp. 1280-1304 ◽  
Author(s):  
Pauline Klein ◽  
Xavier Antoine ◽  
Christophe Besse ◽  
Matthias Ehrhardt
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Absorbing Boundary Conditions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Absorbing Boundary ◽  
One Dimensional ◽  
Nonlinear Potential ◽  
Linear And Nonlinear ◽  
The One

AbstractWe propose a hierarchy of novel absorbing boundary conditions for the one-dimensional stationary Schrödinger equation with general (linear and nonlinear) potential. The accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schrödinger equations. It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition. Finally, we give the extension of these ABCs to N-dimensional stationary Schrödinger equations.

Implementation of fictitious absorbing layers with deceleration effects for one-dimensional Schrödinger equations

2020 ◽  
Vol 86 (5) ◽  
Author(s):  
Hitoshi Kanai ◽  
Tomo Tatsuno
Keyword(s):  
Absorbing Boundary Conditions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Wave Reflections ◽  
Absorbing Boundary ◽  
Dispersive Waves ◽  
One Dimensional ◽  
Phase Velocities ◽  
Absorbing Layer ◽  
And Performance

Absorbing boundary conditions or layers are used in simulations to reduce or eliminate wave reflections from the boundary; one of the most widely used absorbing layers is Berenger's perfectly matched layer (PML). In this paper, PML is extended to a compound absorbing layer which has multiple effects of damping and deceleration, and is applied to linear and nonlinear Schrödinger equations. The deceleration extends the time to damp out the modes with higher phase velocities, leading to remarkably reduced total reflection for dispersive waves. By invoking the two effects independently, the flexibility and performance are enhanced. Since this method is based on the WKB formalism, it requires an absorbing layer of a moderate size.

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