The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

Abstract

We study the new streamline diffusion finite element method for treating the three dimensional coupled nonlinear Schrödinger equation. We derive stability estimates and optimal convergence rates. Moreover, an a priori error estimate is obtained and we compare the corresponding optimal convergence rate for popular numerical methods such as conservative finite difference, semi-implicit finite difference, semi-discrete finite element and the time-splitting spectral method. We justify the advantage of the streamline diffusion method versus the some numerical methods with some examples. Test problems are presented to verify the efficiency and accuracy of the method. The results reveal that the proposed scheme is very effective, convenient and quite accurate for such considered problems rather than other methods.

Stability analysis for some numerical schemes of partial differential equation with extra measurements

“…Various types of physical and environmental phenomena such as heat, electrostatic, electrodynamics, fluid dynamics and pollution could be modelled in the form of partial differential equations (PDEs) [1,2,19,20]. Usually, in study of these phenomena all data of problem is available.…”