TY - JOUR

T1 - Numerical investigations on self-similar solutions of the nonlinear diffusion equation

AU - Li, Yibao

AU - Kim, Junseok

N1 - Funding Information:
The corresponding author (J. Kim) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0023794 ). The authors thank the help of Dr. Youngsoo Ha and Professor Tim Myers in using the ODE solver, bvp5c. The authors greatly appreciate the reviewers for their constructive and insightful comments on this article.

PY - 2013/11

Y1 - 2013/11

N2 - 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.

AB - 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.

KW - Nonlinear multigrid method

KW - Self-similar solution

KW - Thin film

UR - http://www.scopus.com/inward/record.url?scp=84882449695&partnerID=8YFLogxK

U2 - 10.1016/j.euromechflu.2013.05.003

DO - 10.1016/j.euromechflu.2013.05.003

M3 - Article

AN - SCOPUS:84882449695

VL - 42

SP - 30

EP - 36

JO - European Journal of Mechanics, B/Fluids

JF - European Journal of Mechanics, B/Fluids

SN - 0997-7546

ER -