SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Instability of the split-step and related methods near localized solutions of nonlinear Schrödinger equations

T. Lakoba

Abstract
The split-step method (SSM) is widely used for numerical solution of nonlinear evolution equations, e.g., the generalized NLS: $ i u_t + u_{xx} + (V(x)+2|u|^2)u = 0. $ We will show how SSM's instability is analyzed using a modified linearized NLS for a high-wavenumber numerical error. For example, for the SSM which uses the finite-difference solution of its dispersive step, this modified linearized NLS is: $ i\psi_t + \delta\psi - \alpha \psi_{\chi\chi} + V(\epsilon\chi)\psi + 2|u_{\rm sol}(\epsilon\chi)|^2 (2\psi + \psi^*) =0, $ where $\psi$ is proportional to the numerical error. We will show how the instability differs for a stationary, moving, and oscillating background pulse. We will also highlight differences between instabilities of the finite-difference and Fourier implementations of the SSM, and then discuss how these results can be used to predict the instability of the integrating factor and exponential time differencing methods.

Organized by         Universidad de Valladolid     IMUVA