A Comparative Study of Finite Difference and Pseudo-Spectral Methods for a Nonlinear Dynamic Problem

A.T. Jameel (Malaysia) and A. Sharma (India)


Dynamic modeling, thin film flow, Fourier collocation, finite difference


The dynamics of thin film flow on solid plane is represented by equation of evolution (EOE) – a nonlinear fourth order partial differential equation. The extent of nonlinearity and stiffness of the EOE depend upon the nature of the different types of intermolecular and external forces accounted for in the equation of motion, and also to various physico-chemical effects applied on the thin film. Here, we consider a simple model of an ultrathin film of Newtonian liquid on a plane solid substrate subjected to apolar van der Waals and polar intermolecular forces [2, 3]. The EOE was solved numerically for periodic boundary conditions as initial value problem when the free surface of the film was initially perturbed by a sinusoidal wave. Two different classes of numerical methods: a pseudo-spectral and an implicit finite difference schemes were used. The numerical results from the two techniques are compared. It is shown that the Fourier collocation (FC), a pseudo spectral method is easy to implement for nonlinear problems with periodic boundary conditions. The computation time required for the Crank Nicholson, an implicit finite difference scheme (FD) was found to be an order of magnitude larger than that of FC. Thus FC is far more efficient with ease of implementation than FD at least for the problem at hand.

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