scholarly journals Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

2009 ◽  
Vol 43 (4) ◽  
pp. 651-676 ◽  
Author(s):  
François Castella ◽  
Guillaume Dujardin
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Splitting Scheme ◽  
Gevrey Regularity ◽  
Fully Discrete

scholarly journals FULLY DISCRETE SCHEMES FOR THE SCHRÖDINGER EQUATION: DISPERSIVE PROPERTIES

2007 ◽  
Vol 17 (04) ◽  
pp. 567-591 ◽  
Author(s):  
LIVIU I. IGNAT
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Continuous Model ◽  
Local Smoothing ◽  
Euler Approximation ◽  
One Dimensional ◽  
Fully Discrete ◽  
Backward Euler ◽  
The One

We consider fully discrete schemes for the one-dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrödinger equation with nonlinearities which cannot be treated by energy methods.

scholarly journals On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential

2020 ◽  
Vol 54 (5) ◽  
pp. 1491-1508 ◽  
Author(s):  
Chunmei Su ◽  
Xiaofei Zhao
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fixed Time ◽  
Nonlinear Schrodinger Equation ◽  
Splitting Scheme ◽  
Oscillatory Potential ◽  
Nonlinear Schrödinger ◽  
Strang Splitting ◽  
Highly Oscillatory

In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.

scholarly journals On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Saray Busto ◽  
Michael Dumbser ◽  
Cipriano Escalante ◽  
Nicolas Favrie ◽  
Sergey Gavrilyuk
Keyword(s):  
Schrödinger Equation ◽  
Discontinuous Galerkin ◽  
Water Waves ◽  
Schrodinger Equation ◽  
High Order ◽  
First Order ◽  
Fully Discrete ◽  
New Class ◽  
Dispersive Systems ◽  
One Step

AbstractThis paper is on arbitrary high order fully discrete one-step ADER discontinuous Galerkin schemes with subcell finite volume limiters applied to a new class of first order hyperbolic reformulations of nonlinear dispersive systems based on an extended Lagrangian approach introduced by Dhaouadi et al. (Stud Appl Math 207:1–20, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718–2736, 2017). We consider the hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre–Green–Naghdi model of dispersive water waves and the defocusing nonlinear Schrödinger equation. The first order hyperbolic reformulation of the Schrödinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions. We show that the original model proposed in Dhaouadi et al. (2018) is only weakly hyperbolic in the multi-dimensional case and that strong hyperbolicity can be restored at the aid of a novel thermodynamically compatible GLM curl cleaning approach that accounts for the curl involution constraint in the PDE system. We show one and two-dimensional numerical results applied to both systems and compare them with available exact, numerical and experimental reference solutions whenever possible.

Fully Discrete Galerkin Finite Element Method for the Cubic Nonlinear Schrödinger Equation

2017 ◽  
Vol 10 (3) ◽  
pp. 671-688 ◽  
Author(s):  
Jianyun Wang ◽  
Yunqing Huang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Fully Discrete ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Element Method

AbstractThis paper is concerned with numerical method for a two-dimensional time-dependent cubic nonlinear Schrödinger equation. The approximations are obtained by the Galerkin finite element method in space in conjunction with the backward Euler method and the Crank-Nicolson method in time, respectively. We prove optimalL2error estimates for two fully discrete schemes by using elliptic projection operator. Finally, a numerical example is provided to verify our theoretical results.

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