Title: Time-dependent spectral renormalization method,
Abstract: The spectral renormalization method was introduced by Ablowitz and Musslimani
in 2005, [Opt. Lett. 30, pp. 2140-2142] as an effective way to numerically
compute (time-independent) bound states for certain nonlinear boundary value
problems. % of the nonlinear Schrödinger (NLS), Gross-Pitaevskii and water
wave type equations to mention a few. In this paper, we extend those ideas to
the time domain and introduce a time-dependent spectral renormalization method
as a numerical means to simulate linear and nonlinear evolution equations. The
essence of the method is to convert the underlying evolution equation from its
partial or ordinary differential form (using Duhamel's principle) into an
integral equation. The solution sought is then viewed as a fixed point in both
space and time. The resulting integral equation is then numerically solved
using a simple renormalized fixed-point iteration method. Convergence is
achieved by introducing a time-dependent renormalization factor which is
numerically computed from the physical properties of the governing evolution
equation. The proposed method has the ability to incorporate physics into the
simulations in the form of conservation laws or dissipation rates. This novel
scheme is implemented on benchmark evolution equations: the classical nonlinear
Schrödinger (NLS), integrable $PT$ symmetric nonlocal NLS and the viscous
Burgers' equations, each of which being a prototypical example of a
conservative and dissipative dynamical system. Numerical implementation and
algorithm performance are also discussed.