The Spectral Approximation For Coupled Nonlinear Schr(o|¨)dinger Equations

It’s known to all that Coupled Nonlinear Schrodinger (CNLS) equatios have a very wide range of applications in the areas of science and engineering. In this pa-per, we considered the numerical solution of CNLS equations by orthogonal spline collocation method and the Legendre spectral approximation method.Specifically, firstly we used orthogonal spline collocation method in spatial direction and stable second-order implicit scheme in the time direction. And we proved the conservation properties by using the energy method.Then we strictly proved its second-order accuracy in the time direction and high-order accuracy in the spatial direction. Secondly, in order to obtain higher accuracy in the spatial direction,a full implicit second-order Legendre Spectrum is presented. The conservation law and accura-cy of its numerical solution is rigorously proved by the discrete energy method and then we gave an estimate of the convergence order. Finally,we presented the effectiveness of the method by the numerical experiments.