Crank-Nicolson/finite element approximation for the Schrödinger equation in the de Sitter spacetime

Abstract

Central to much of science, engineering, and society today is the building of mathematical models to represent complex processes. Recently, the non-relativistic limit of nonlinear Klein-Gordon equations in de Sitter spacetime has been used to derive Schrödinger equations with weighted nonlinear terms In this paper, numerical simulations are constructed to clarify the behavior of the solution in both one- and two-dimensions. These simulations are constructed based on the Crank-Nicolson scheme in the time direction and the Galerkin finite element in the spatial direction. The nonlinear system of algebraic equations resulting from the constructed scheme is solved using Newton’s method. It is also demonstrated that the numerical approximation converges to the exact one.

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

“…There are various results concerning the numerical simulation of the nonlinear Schrödinger equation and its variants. Such results include finite difference method, [1][2][3][4][5][6] finite element method, [7][8][9][10] spectral collocation methods, [11][12][13][14][15] and meshless technique. [16][17][18][19] To the best of our knowledge, there have only been a few results on the numerical solution of the Schrödinger equation in unbounded domain.…”

“…There are various results concerning the numerical simulation of the nonlinear Schrödinger equation and its variants. Such results include finite difference method, 1–6 finite element method, 7–10 spectral collocation methods, 11–15 and meshless technique 16–19 …”