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

Yibao Li, Junseok Kim

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


In this paper, we present the numerical investigations of self-similar solutions for the nonlinear diffusion equation ht=-(h 3hxxx)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.

Original languageEnglish
Pages (from-to)30-36
Number of pages7
JournalEuropean Journal of Mechanics, B/Fluids
Publication statusPublished - 2013 Nov


  • Nonlinear multigrid method
  • Self-similar solution
  • Thin film

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)


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