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 h_t=-(h ³h_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(α(t)(x-x ₀)+ x₀, t₀)=f(α(t)(x-x₀)) and α(t)=[1-4A(t-t₀)]^-1/4, 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, x ₀, 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.