Nonlinear Sciences > Pattern Formation and Solitons
[Submitted on 22 Feb 2017 (v1), last revised 2 Aug 2017 (this version, v2)]
Title:Time-dependent spectral renormalization method
View PDFAbstract: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.
Submission history
From: Ziad Musslimani [view email][v1] Wed, 22 Feb 2017 15:35:07 UTC (407 KB)
[v2] Wed, 2 Aug 2017 14:12:39 UTC (696 KB)
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