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arXiv · 1907.07082

A Reduced Order Modeling technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation

Federico Pichi·Annalisa Quaini·Gianluigi Rozza
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Abstract

We propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called Gross-Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.

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Federico Pichi, Annalisa Quaini, Gianluigi Rozza. 2020-06-10. A Reduced Order Modeling technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation. https://arxiv.org/abs/1907.07082

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