A Krylov semi-implicit discontinuous Galerkin method for the computation of ground and excited states in Bose–Einstein condensates

• The discontinuous Galerkin method to capture the interior layers in excited states.
• A Krylov approximation is used to compute the matrix-vector products.
• The semi-implicit DG method is stable and efficient.

Numerical computations of ground and excited states of Bose–Einstein condensates (BECs) require high spatial resolution due to the presence of the multiscale structures and boundary layers and interior layers in the solution. In this paper, a new discontinuous Galerkin (DG) method is presented for the computation of ground and excited states of BECs. For the spatial discretization, the direct discontinuous Galerkin (DDG) method is applied by using the normalized gradient flow. For the time discretization, we integrate the ordinary differential equations (ODEs) which is obtained by linearizing the interaction term. To evaluate the matrix exponential operator efficiently, we apply the Krylov subspace approximations to the matrix exponential operator. Numerical examples with different potentials are reported to demonstrate the validity and effectiveness of the semi-implicit DG method.