| The nonlinear Schro(o|¨)dinger equation (NLS) is a kind of important nonlinear evolutionequation. It has important applications in plasma physics, nonlinear optics, superconductor,and has been investigated in many ways. These investigations are focused on the develop-ment of numerical methods, most of which are the finite difference and finite element meth-ods[1–4]. A few works have been done on the Fourier spectral method and pseudo-spectralmethod to solve the NLS equation with periodic boundary by many authors[5]. However,there is a need to consider the non-periodic problem. Our goal is to develop the finite differ-ence in time and spectral approximation in space for the NLS equation with Dirichlet bound-ary. Since the Fourier spectral method is no longer suitable for the non-periodic problem, weturn to the Legendre spectral method.Precisely, we consider the initial- and Dirichlet boundary-value problem for a class ofthe NLS equation with power nonlinear term. The presence of the nonlinear term makes theconstruction of the conservative scheme difficult. Based on second-order implicit differencescheme in time and Legendre spectral element method in space, we develop an conservativeand efficient scheme to the NLS equation. For the semi-discrete approximation, we prove theconservative properties and give the L2 error estimate. For the full-discrete approximation,we study the existence and uniqueness of the solution by using the fixed-point theory, andprove the L2 error bound of optimal order of accuracy. Finally, some numerical experimentsare performed to support our theoretical claims.
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