Numerical investigations on self-similar solutions of the nonlinear diffusion equation

• We describe a numerical method for calculating self-similar solutions of this film.
• We perform several numerical tests to demonstrate that the numerical simulations are in qualitative agreement with self-similar solutions.
• Various numerical experiments are performed to show that the proposed algorithm can generate a self-similar solution.

In this paper, we present the numerical investigations of self-similar solutions for the nonlinear diffusion equation ht=−(h3hxxx)xht=−(h3hxxx)x, which arises in the context of surface-tension-driven flow of a thin viscous liquid film. Here, h=h(x,t)h=h(x,t) is the liquid film height. A self-similar solution is h(x,t)=h(α(t)(x−x0)+x0,t0)=f(α(t)(x−x0))h(x,t)=h(α(t)(x−x0)+x0,t0)=f(α(t)(x−x0)) and α(t)=[1−4A(t−t0)]−1/4α(t)=[1−4A(t−t0)]−1/4, where AA and x0x0 are constants and t0t0 is a reference time. To discretize the governing equation, we use the Crank–Nicolson finite difference method, which is second-order accurate in time and space. The resulting discrete system of equations is solved by a nonlinear multigrid method. We also present efficient and accurate numerical algorithms for calculating the constants, AA, x0x0, and t0t0. To find a self-similar solution for the equation, we numerically solve the partial differential equation with a simple step-function-like initial condition until the solution reaches the reference time t0t0. Then, we take h(x,t0)h(x,t0) as the self-similar solution f(x)f(x). Various numerical experiments are performed to show that f(x)f(x) is indeed a self-similar solution.