An efficient numerical algorithm for the solution of nonlinear problems with periodic boundary conditions

A nonlinear fourth order partial differential equation representing the stability of a nano-thin film of Newtonian liquid on solid plane was solved numerically for periodic boundary conditions and a sine wave initial condition imposed on the free surface of the film, using a pseudo-spectral and an implicit finite difference method. The numerical results from the two 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 in the case of FC. Thus the FC is far more efficient than FD at least for the nonlinear periodic problem at hand.