低正则位势与非线性下非线性薛定谔方程指数波积分器的最优误差界
Optimal error bounds on the exponential wave integrator for the nonlinear Schrödinger equation with low regularity potential and nonlinearity
AI总结：
本文针对低正则位势与非线性下的非线性薛定谔方程，建立了指数波积分器在时间半离散和 Fourier 谱全离散下的最优 L^2 与 H^1 误差界，并提出扩展 Fourier 拟谱方法以降低计算代价，数值实验验证了误差界的精确性。
摘要：
我们针对带有 L^∞-位势和/或局部 Lipschitz 非线性的非线性薛定谔方程（NLSE），在方程具有 H^2-解的假设下，建立了指数波积分器（EWI）的最优误差界。对于时间上采用一阶 Gautschi 型 EWI 的半离散格式，我们证明了最优 L^2-误差界为 O(τ)，其中 τ>0 为时间步长，同时证明了数值解的一致 H^2-界。对于空间上采用 Fourier 谱方法得到的全离散格式，我们证明了最优 L^2-误差界为 O(τ+h^2)，且 τ 与 h 之间无需任何耦合条件，其中 h>0 为网格尺寸。此外，对于 W^{1,4}-位势以及稍强的非线性正则性，在 H^3-解的假设下，我们得到了最优 H^1-误差界。进一步地，当位势正则性较低但非线性足够光滑时，我们提出了一种扩展的 Fourier 拟谱方法，该方法具有与 Fourier 谱方法相同的误差界，而其计算代价与标准 Fourier 拟谱方法相近。我们的新误差界极大地改进了现有关于低正则位势和/或非线性 NLSE 的结果。我们报告了大量数值结果以验证我们的误差估计，并证明这些误差界是精确的。
英文摘要：
We establish optimal error bounds for the exponential wave integrator (EWI) applied to the nonlinear Schrödinger equation (NLSE) with $ L^\infty $-potential and/or locally Lipschitz nonlinearity under the assumption of $ H^2 $-solution of the NLSE. For the semi-discretization in time by the first-order Gautschi-type EWI, we prove an optimal $ L^2 $-error bound at $ O(τ) $ with $ τ>0 $ being the time step size, together with a uniform $ H^2 $-bound of the numerical solution. For the full-discretization scheme obtained by using the Fourier spectral method in space, we prove an optimal $ L^2 $-error bound at $ O(τ+ h^2) $ without any coupling condition between $ τ$ and $ h $, where $ h>0 $ is the mesh size. In addition, for $ W^{1, 4} $-potential and a little stronger regularity of the nonlinearity, under the assumption of $ H^3 $-solution, we obtain an optimal $ H^1 $-error bound. Furthermore, when the potential is of low regularity but the nonlinearity is sufficiently smooth, we propose an extended Fourier pseudospectral method which has the same error bound as the Fourier spectral method while its computational cost is similar to the standard Fourier pseudospectral method. Our new error bounds greatly improve the existing results for the NLSE with low regularity potential and/or nonlinearity. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.
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