Long Time Dynamics Of Implicit Euler Method For The Allen-Cahn Equation

The Allen-Cahn(A-C)equation is the primary model simulating phase-field in materials science,which describes the process of a binary alloys at a certain temperature phase separation.In this paper,we consider the initial value problem for the Allen-Cahn equation with Nuemann boundary,use implicit Euler method to discretize the time variables,and get the following results.1.First,we proved the important property of this equation which still preserves the property of energy decay in discrete sense,ε(un+1)=∫Ω(1/2|▽un+1|2+1/ε2F(un+1))dx≤ε(un).2.Next,the uniformly dissipative estimate of implicit Euler scheme on spatial L2 is obtained,‖un‖≤ρ,tn≥tn0.where,the constant ρ>ρ0k=4γ2C1(α),and is an positive integer tn0=tn0(R,α,ρ,k).3.Again,the uniformly dissipative estiaate of implicit Euler scheme on space Hl is obtained,‖▽un+1‖≤ρ1k,ρ1k=(2C1(α)+ρ0k2/r)1/2.4.Finally,We use numerical experiments to verify the theory and firther illustrate the long-term stability of the implicit Euler scheme for the Allen-Cahn equation.