Numerical Analysis Of Cahn-Hilliard Equation And Cahn-Hilliard-Brinkman Equation

Cahn-Hilliard equation is a very important fourth order nonlinear parabolic partial differential equation.It is used to describe the phase separation process and coarsening phenomenon of binary alloys in some unstable state.At present,most of the numerical methods are first order precision.The second order accuracy numerical format is better than the first order numerical format in terms of numerical efficiency and accuracy,but the analysis of the second order format is more difficult than that of the first order format.The second order numerical scheme of Cahn-Hilliard equation and Cahn-Hilliard-Brinkman equation is analyzed.For the first part,the standard second-order convex split Crank-Nicolson scheme and the backward difference formula scheme(hereafter referred to as the BDF scheme)are extended in time.The second order precision format of Cahn-Hilliard equation with ?(????1/2)parameter is presented,when ?=0 is the second-order convex split Crank-Nicolson scheme,and when ?=1/2 is the second-order convex split scheme,where polynomial potential functions use concave-convex splitting.In order to ensure the energy stability,Dougla-Dupont regularization term A?(1-2?)?(?hn-?hn-1)is added.It satisfyies the condition A?1/16.The finite element method is used to discretize the space,which proves that the scheme is energy attenuation.Then the error analysis is carried out.Finally numerical experiments show that the proposed format is second order accuracy in time.In the second part,the numerical scheme of the variable coefficient Cahn-Hilliard-Brinkman equation is given.The numerical scheme uses finite element method in space and energy convex splitting method in time.It is proved that the dispersion scheme is energy attenuated.The term containing concentration and Peclet number is decom-posed into two terms by using the Cauchy median theorem in error analysis.The results show that the proposed format is second order accuracy in time.