Stability and convergence of Dufort-Frankel-type difference schemes for a nonlinear Schrödinger-type equation (Q1386198)

The first boundary value problem for the nonlinear Schrödinger equation \[ {\partial u\over\partial t}= ia{\partial^2u\over\partial x^2}+ f(u, u^*)u, \] which appears in nonlinear optic models to describe an energy transfer in molecular systems and also in quantum mechanics, seismology, plasma physics, theories of vortex motion and superconductivity, and other domains of natural science is investigated. The convergence of a three-layer explicit difference scheme for the numerical approximation of this problem in the function spaces \(C\) and \(W^1_2\) is proved. For justification of convergence and stability the grid analogues of energy preservation laws and grid multiplicative inequalities are used. For the grid stepsize the relation \(2| a|\tau/h^2\leq \nu<1\) is assumed.