A Taylor-Galerkin method for the use of finite elements in a nonlinear problem for the numerical modeling of a new Chemo-Fluid oscillator

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Illych Álvarez, David Terán, Danny Dessechis

Abstract

In this work, a finite element scheme is proposed using a method of Euler-Taylor-Galerkin
described inPáez (2016), for a non-linear model which describes the behavior of a new chemofluidic
oscillator (Donea, 1984). This model is expressed by the coupling of an ordinary
differential equation describing the hydrogel dynamics, the non-linear transport equation and an
auxiliary equation determining the flux volume. The numerical solution is constructed by taking a
semi-discretization in time of the transport equation, employing forward-time Taylor series
expansions including time derivatives of second order and third order, avoiding instabilities
problems. In this semi discrete equation, the spatial variable is approximated by the finite element
formulation according to Galerkin. Some simulations are carried out taking different initial
conditions for the concentration of the hydrogel. The numerical results describe the oscillatory
behavior of the system as in Donea (1984), where MatLab tools are used as black box.

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