This article introduces a new numerical method for the minimization under constraints of a discrete energy modeling multicomponents rotating Bose-Einstein condensates in the regime of strong confinement and with rotation. Moreover, we consider both segregation and coexistence regimes between the components. The method includes a discretization of a continuous energy in space dimension 2 and a gradient algorithm with adaptive time step and projection for the minimization. It is well known that, depending on the regime, the minimizers may display different structures, sometimes with vorticity (from singly quantized vortices, to vortex sheets and giant holes). In order to study numerically the structures of the minimizers, we introduce in this paper a numerical algorithm for the computation of the indices of the vortices, as well as an algorithm for the computation of the indices of vortex sheets. Several computations are carried out, to illustrate the efficiency of the method, to cover different physical cases, to validate recent theoretical results as well as to support conjectures. Moreover, we compare this method with an alternative method from the literature.

翻译：本文提出了一种新的数值方法，用于在强约束和旋转条件下，对描述多组分旋转玻色-爱因斯坦凝聚体的离散能量进行约束最小化。此外，我们同时考虑了组分间的分离态和共存态。该方法包括空间二维连续能量的离散化，以及采用自适应时间步长和投影的梯度算法进行最小化。众所周知，根据状态的不同，极小化子可能呈现不同的结构，有时伴有涡度（从单量子化涡旋到涡旋片和巨洞）。为了数值研究极小化子的结构，本文提出了一种计算涡旋指数的数值算法，以及一种计算涡旋片指数的算法。我们进行了多项计算，以说明该方法的有效性，涵盖不同的物理情形，验证近期理论结果并支持相关猜想。此外，我们将该方法与文献中的替代方法进行了比较。