Stabilizing the nonlinear initial boundary value problem governing thin film flow

Abstract

We study the nonlinear initial boundary value problem (IBVP) and that governs thin film flows. The generally accepted version is shown not to lead to an energy bound for all parameter values. Firstly, we introduce modifications to the continuous IBVP. The first modification retains the original thin film flow equation but relates the allowed parameter value to the first derivative of the solution. The second modification relates the allowed parameter value to the second derivative of the solution which slightly modifies the diffusive flux. Secondly, we develop new splitting techniques for the nonlinear terms. These new splitting techniques are essential for stability of the semi-discretised IBVP and are applied to both the nonlinear advection terms and the highly non-linear diffusive terms. Thirdly, we implement boundary conditions weakly that together with the development described above leads to provably conservative and stable nonlinear schemes. Numerical calculations corroborate the developments, exemplify the influence of the modifications, and reveal important parameter dependencies.