Scalar Auxiliary Variable/Lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations

Abstract

In this paper, based on the Scalar Auxiliary Variable (SAV) approach [44,45] and a newly proposed Lagrange multiplier (LagM) approach [22,21] originally constructed for gradient flows, we propose two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrödinger/Gross-Pitaevskii equations. Both schemes are of spectral/second-order accuracy in spatial/temporal direction. The SAV based scheme preserves a modified total energy and approximate the mass to third order (with respect to time steps), while the LagM based scheme could preserve exactly the mass and original total energy. A nonlinear algebraic system has to be solved at every time step for the LagM based scheme, hence the SAV scheme is usually more efficient than the LagM one. On the other hand, the LagM scheme may outperform the SAV ones in the sense that it conserves the original total energy and mass and usually admits smaller errors. Ample numerical results are presented to show the effectiveness, accuracy and performance of the proposed schemes.