ACCURATE SOLUTION OF THE NONLINEAR SCHRÖDINGER EQUATION VIA CONSERVATIVE MULTIPLE-RELAXATION IMEX METHODS

The nonlinear Schrödinger (NLS) equation possesses an infinite hierarchy of conserved densities, and the numerical preservation of some of these quantities is critical for accurate long-time simulations. We propose a discretization that conserves one or two of these conserved quantities by combining higher-order implicit-explicit (ImEx) Runge-Kutta time integrators with the relaxation technique and adaptive step size control and only requires the solution of one or two algebraic equations at the end of each step. We show through numerical tests that our mass-conserving method is much more efficient and accurate than the widely used second-order time-splitting pseudospectral approach. Compared to higher-order operator splitting, it gives similar results in general and significantly better results near the semiclassical limit. Furthermore, for some problems adaptive time stepping provides a dramatic reduction in cost without sacrificing accuracy. We also propose a full discretization that conserves both mass and energy by using a conservative finite element spatial discretization and multiple relaxation in time. Our results suggest that this method provides a qualitative improvement in long-time error growth for multi-soliton solutions.