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

本文对一维非线性 Schrödinger 方程给出两个紧致差分格式, 运用能量方法和两个新的分析技 巧证明格式关于离散质量和离散能量守恒, 而且在最大模意义下无条件收敛. 对非线性紧格式构造了 一个新的迭代算法, 证明了算法的收敛性, 并在此基础上给出一个新的线性化紧格式. 数值算例验证 了理论分析的正确性, 并通过外推进一步提高了数值解的精度.

Unconditional convergence of two conservative compact difference schemes for non-linear Schrodinger equation in one dimension

Two compact difference schemes for one-dimensional cubic nonlinear SchrSdinger equation are proposed. 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 generated 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. Numerical examples are given to support the theoretical analysis, and two Richardson extrapolations are introduced to achieve higher order numerical accuracy.