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

In this paper, we present the numerical investigations of self-similar solutions for the nonlinear diffusion equation ht = −(hhxxx)x, which arises in the context of surface-tension-driven flow of a thin viscous liquid film. Here, 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)) and α(t) = [1 − 4A(t − t0)], where A and x0 are constants and t0 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, A, x0, and t0. 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 t0. Then, we take h(x, t0) as the self-similar solution f (x). Various numerical experiments are performed to show that f (x) is indeed a self-similar solution. © 2013 Elsevier Masson SAS. All rights reserved.