一维非线性Schrödinger 方程的两个无条件收敛的守恒紧致差分格式
摘要
Two compact difference schemes for one-dimensional cubic nonlinear Schrödinger equation are pro- posed. Both of the schemes are proved to conserve the discrete mass and energy. Unconditional convergence in maximum norm of the difference solutions with forth order in space and second order in time is also proved by utilizing the discrete energy method and two techniques. For computing the nonlinear algebraical system gener- ated by the nonlinear compact scheme, an iterative algorithm is constructed and proved to be convergent. As a byproduct of the iterative algorithm, a linearized compact difference scheme with high accuracy is obtained. Nu- merical examples are given to support the theoretical analysis, and two Richardson extrapolations are introduced to achieve higher order numerical accuracy.