求解耗散Schrdinger方程的一个无条件收敛的线性化紧致差分格式

An Unconditionally Convergent and Linearized Compact Finite Difference Scheme for the Nonlinear Schrdinger Equation with a Dissipative Term

本文致力于提出并分析一个求解耗散Schrdinger方程的线性化紧致差分格式.通过引入—个新的变量来消除耗散项,原方程可化为一个保持总质量和总能量的守恒系统.本文继而对这个守恒系统提出了一个高效的紧致差分格式,并证明该格式在离散意义下保持总质量和总能量守恒.运用不动点定理和标准的能量方法,新格式被证明是唯一可解的.不同于经典的基于数值解先验估计的分析方法,本文引进数学归纳法并结合H^1估计,在对网格比没有任何要求的前提下建立了格式在最大模意义下的最优误差估计.格式的收敛阶在空间和时间两个方向分别为4阶和2阶.数值结果验证了理论分析的正确性,并展示了新格式较已有格式的优越性.

This paper aims to design and analyze a linearized compact finite difference scheme for solving a dissipative nonlinear Schrodinger equation. By introducing a transform of variable to eliminate the dissipative term, a new system which preserves the total mass and energy is obtained. Then an efficient compact finite difference scheme is proposed for the new system. Corresponding to the fact that the original problem preserves the total mass and energy, the novel scheme is proved to also preserve the two invariant quantities in the discrete sense. The unique solvability of the numerical scheme is proved by using a fixed point theorem together with the standard energy method. Unlike the classical analysis method based on the a priori estimate of the numerical solution, we here introduce an induction argument together with an H^1 technique in building the optimal error bound. The convergence rate is proved to be of fourth-order in space and second-order in time, respectively. Numerical results are reported to support the theoretical analysis and show that the novel scheme is superior to the existing scheme.