A three operator split-step method covering a larger set of non-linear partial 
differential equations

Zia, Haider

doi:10.1016/j.cnsns.2016.11.020

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A three operator split-step method covering a larger set of non-linear partial differential equations

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Zia, Haider
Miller Group, Atomically Resolved Dynamics Department, Max Planck Institute for the Structure and Dynamics of Matter, Max Planck Society;

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http://arxiv.org/abs/1604.01138
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https://dx.doi.org/10.1016/j.cnsns.2016.11.020
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Citation

Zia, H. (2017). A three operator split-step method covering a larger set of non-linear partial differential equations. Communications in Nonlinear Science and Numerical Simulation, 47, 277-291. doi:10.1016/j.cnsns.2016.11.020.

Cite as: https://hdl.handle.net/11858/00-001M-0000-002A-5F34-6

Abstract

This paper describes an updated Fourier based split-step method that can be applied to a greater class of partial differential equations, than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized non-linear Schr\"odinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.