Accurate Solution of the Nonlinear Schrödinger Equation via Conservative Multiple-Relaxation ImEx Methods

by   Abhijit Biswas, et al.
King Abdullah University of Science and Technology

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, particularly for multi-soliton solutions. We propose an essentially explicit 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. We show through numerical tests that our mass-conserving method is much more efficient and accurate than the widely-used 2nd-order time-splitting pseudospectral approach. Compared to higher-order operator splitting, it gives similar results in general and significantly better results near the semi-classical 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.


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