The time-dependent Gross-Pitaevksii equation (GPE) is a nonlinear Schr\"odinger equation which is used in quantum physics to model the dynamics of Bose-Einstein condensates. In this work we consider numerical approximations of the GPE based on a multiscale approach known as the localized orthogonal decomposition. Combined with an energy preserving time integrator one derives a method which is of high order in space and time under mild regularity assumptions. In previous work, the method has been shown to be numerically very efficient compared to first order Lagrange FEM. In this paper, we further investigate the performance of the method and compare it with higher order Lagrange FEM. For rough problems we observe that the novel method performs very efficient and retains its high order, while the classical methods can only compete well for smooth problems.

翻译：含时Gross-Pitaevskii方程（GPE）是一种非线性薛定谔方程，在量子物理学中用于模拟玻色-爱因斯坦凝聚体的动力学。本文基于一种称为局部正交分解的多尺度方法，研究GPE的数值逼近。结合能量守恒时间积分器，该方法在温和正则性假设下，可在空间和时间维度上达到高阶精度。先前研究表明，与一阶拉格朗日有限元法相比，该方法具有极高的数值效率。本文进一步考察了该方法的性能，并将其与高阶拉格朗日有限元法进行对比。对于粗糙问题，我们观察到新方法仍保持高效性与高阶精度，而经典方法仅能良好地求解光滑问题。