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

Abstract

In this paper, we present the numerical investigations of self-similar solutions for the nonlinear diffusion equation h(t) = -(h(3)h(xxx))(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(alpha(t)(x - x(0)) + x(0), t(0)) = f(alpha(t)(x - x(0))) and alpha(t) = [1 - 4A(t - t(0))](-1/4), where A and x(0) are constants and t(0) 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, x(0), and t(0). 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 to. Then, we take h(x, t(0)) as the self-similar solution f(x). Various numerical experiments are performed to show that f(x) is indeed a self-similar solution. (C) 2013 Elsevier Masson SAS. All rights reserved.