Operator-compensation methods with mass and energy conservation for solving the Gross-Pitaevskii equation

• Published in: Applied numerical mathematics Vol. 151; pp. 337 - 353
• Main Authors: Li, Xiang-Gui, Cai, Yongyong, Wang, Pengde
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.05.2020
• Subjects: Conservation; Gross-Pitaevskii equation; High-order methods; Operator-compensation methods; Operator-compensation methods; High-order methods; Gross-Pitaevskii equation; Conservation
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2020.01.004

In this work, operator-compensation based high order methods are presented to solve the Gross-Pitaevskii equation modeling Bose-Einstein condensation (BEC). We begin with the high-order approximation to the Laplacian by the central finite difference method combined with operator-compensation technique, and the Crank-Nicolson and time-splitting methods are then proposed for the time discretization. We show that the Crank-Nicolson operator-compensation method keeps the mass and energy conservation, while the time-splitting operator-compensation method only conserves the mass. Then by extending the high-order operator-compensation methods for the first order derivative, the time-splitting finite difference method is developed to solve the Gross-Pitaevskii equation with an angular momentum rotation term. Numerical results are given to test accuracy and verify the conservation properties.