A level-set based finite difference method for the ground state Bose-Einstein condensates in smooth bounded domains

Abstract: We present a level-set based finite difference method to calculate the ground states of Bose Einstein condensates in domains with curved boundaries. Our method draws on the variational and level set approaches, benefiting from both of their long-standing success. More specifically, we use the normalized gradient flow, where the spatial discretization is based on the simple Cartesian grid with fictitious values in the outer vicinity of the domains. We develop a PDE-based extension technique that systematically and automatically constructs ghost point values with third-order accuracy near irregular boundaries, effectively circumventing the computational complexity of interpolation in these regions. Another novel aspect of our work is the application of the PDE-based extension technique to a nodal basis function, resulting in an explicit ghost value mapping that can be seamlessly incorporated into implicit time-stepping methods where the extended function values are treated as unknowns at the next time step. We present numerical examples to demonstrate the effectiveness of our method, including its application to domains with corners and to problems involving higher-order interaction terms.