Error estimation of the Relaxation Finite Difference Scheme for the nonlinear Schrödinger Equation

by   Georgios E. Zouraris, et al.

We consider an initial- and boundary- value problem for the nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in the one space dimension case. We discretize the problem in space by a central finite difference method and in time by the Relaxation Scheme proposed by C. Besse [C. R. Acad. Sci. Paris Sér. I 326 (1998), 1427-1432]. We provide optimal order error estimates, in the discrete L_t^∞(H_x^1) norm, for the approximation error at the time nodes and at the intermediate time nodes. In the context of the nonlinear Schrödinger equation, it is the first time that the derivation of an error estimate, for a fully discrete method based on the Relaxation Scheme, is completely addressed.



page 1

page 2

page 3

page 4


A Relaxation/Finite Difference discretization of a 2D Semilinear Heat Equation over a rectangular domain

We consider an initial and Dirichlet boundary value problem for a semili...

Optimal error estimate for a space-time discretization for incompressible generalized Newtonian fluids: The Dirichlet problem

In this paper we prove optimal error estimates for solutions with natura...

A Convergent Quadrature Based Method For The Monge-Ampère Equation

We introduce an integral representation of the Monge-Ampère equation, wh...

Community detection in networks via nonlinear modularity eigenvectors

Revealing a community structure in a network or dataset is a central pro...

A derivation of Griffith functionals from discrete finite-difference models

We analyze a finite-difference approximation of a functional of Ambrosio...

On compact 4th order finite-difference schemes for the wave equation

We consider compact finite-difference schemes of the 4th approximation o...

The linearization methods as a basis to derive the relaxation and the shooting methods

This chapter investigates numerical solution of nonlinear two-point boun...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.