2023/09/05 by Abhijit Biswas, Biswas, Abhijit, David I. Ketcheson +1 · 2 citations
Mathematics · Earth and Planetary Sciences · Physics and Astronomy · #Numerical methods for differential equations #Meteorological Phenomena and Simulations #Advanced Fiber Laser Technologies
paper · pdf · doi:10.48550/arxiv.2309.02324
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.