Numerical investigation of a third-order ODE from thin film flow

We compare two finite difference schemes to solve the third-order ordinary differential equation y^‴ = y^-k from thin film flow. The boundary conditions come from Tanner's problem for the surface tension driven flow of a thin film. We show that a central difference approximation to the third derivative in the model equation produces a solution curve with oscillations. A difference scheme based on a combination of forward and backward differences produces a smooth accurate solution curve. Both the 0-stability and von Neumann stability properties of the different finite difference schemes are analyzed. The solution curves obtained from both approaches are presented and discussed.