Finite difference approximation for nonlinear Schr\" odinger equations with application to blow-up computation
Finite difference approximation for nonlinear Schr\" odinger equations with application to blow-up computation
复制标题
非线性Schr" odinger方程的有限差分近似及其在爆破计算中的应用
DOI：
10.1007/s13160-016-0218-8
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
N. Saito and T. Sasaki
中科院分区：
文献类型：
--
作者：
N. Saito and T. Sasaki
本文对一维非线性Schrödinger （NLS）方程的有限差分法进行了相干分析。我们使用离散框架来建立规范中的适定性和误差估计。假设NLS方程的非线性f(u)只满足一个生长条件。将所得结果应用于一类非线性为正实数的NLS方程爆破解的计算。特别地，我们给出了数值爆破时间，其中包含了空间变量和时间变量的离散参数。证明了它收敛于原NLS方程解的爆破时间。通过数值算例验证了理论结果的有效性。此外，我们从数值研究中推断，如果将Crank-Nicolson格式应用于时间离散，则其收敛速度为二阶。
This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discreteframework to establish well-posedness and error estimates in thenorm. The nonlinearityf(u) of a NLS equation is assumed to satisfy only a growth condition. We apply our results to computation of blow-up solutions for a NLS equation with the nonlinearity,pbeing a positive real number. Particularly, we offer the numerical blow-up time, wherehandare discretization parameters of space and time variables. We prove thatconverges to the blow-up timeof the solution of the original NLS equation. Several numerical examples are presented to confirm the validity of theoretical results. Furthermore, we infer from numerical investigation that the convergence ofis at a second order rate inif the Crank–Nicolson scheme is applied to time discretization.