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首页> 外文期刊>SIAM Journal on Numerical Analysis >NUMERICAL DISPERSIVE SCHEMES FOR THE NONLINEAR SCHRODINGER EQUATION
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NUMERICAL DISPERSIVE SCHEMES FOR THE NONLINEAR SCHRODINGER EQUATION

机译：非线性Schrodinger方程的数值色散格式。

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摘要

We consider semidiscrete approximation schemes for the linear Schrodinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semidiscretization scheme we show that, as the mesh size tends to zero, the semidiscrete approximate solutions lose the dispersion property. This fact is proved by constructing solutions concentrated at the points of the spectrum where the second order derivatives of the symbol of the discrete Laplacian vanish. Therefore this phenomenon is due to the presence of numerical spurious high frequencies. To recover the dispersive properties of the solutions at the discrete level, we introduce two numerical remedies: Fourier filtering and a two-grid preconditioner. For each of them we prove Strichartz-like estimates and a local space smoothing effect, uniform in the mesh size. The methods we employ are based on classical estimates for oscillatory integrals. These estimates allow us to treat nonlinear problems with L-2-initial data, without additional regularity hypotheses. We prove the convergence of the two-grid method for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates.
机译：我们考虑线性Schrodinger方程的半离散逼近方案，并分析连续模型的经典色散特性是否适用于这些逼近。对于保守的有限差分半离散方案，我们表明，随着网格尺寸趋于零，半离散近似解会失去色散特性。通过构造集中在不连续拉普拉斯符号的二阶导数消失的频谱点的解决方案，可以证明这一事实。因此，此现象是由于存在数字杂散高频。为了恢复离散级解的分散性，我们引入了两种数值方法：傅立叶滤波和两网格预处理器。对于它们中的每一个，我们都证明了类似于Strichartz的估计值和局部空间平滑效果，并且网格尺寸均匀。我们采用的方法基于振荡积分的经典估计。这些估计值使我们能够使用L-2-初始数据来处理非线性问题，而无需其他正则性假设。我们证明了非线性的两网格方法的收敛性，该方法无法通过能量参数来处理，并且即使在连续情况下，也需要进行Strichartz估计。

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