Long-Time Simulations of Nonlinear Schrödinger-Type Equations using Step Size Exceeding Threshold of Numerical Instability

Abstract

We propose an exponential time-differencing method based on the leapfrog scheme for numerical integration of the generalized nonlinear Schrodinger-type equations. The key advantage of the proposed method over the widely used Fourier split-step method is that in the new method, numerical instability at high wavenumbers is strongly suppressed. This allows one to use time steps that considerably exceed the instability threshold, which leads to a proportional reduction of the computational time. Moreover, we introduce a technique that eliminates numerical instability at low-to-moderate wavenumbers that is common for methods based on the leapfrog scheme. We illustrate the performance of the proposed method with examples from two applications areas: deterministic wave turbulence and solitary waves.