Abstract
We consider in this paper the stabilized semi-implicit (in time) scheme and the splitting scheme for the Allen-Cahn equation $\phi_t-\Delta\phi+$e$^-2f(\phi)=0$ arising from phase transitions in material science. For the stabilized first-order scheme, we show that it is unconditionally stable and the error bound depends on e-1 in some lower polynomial order using the spectrum estimate of [2, 10, 11]. In addition, the first- and second-order operator splitting schemes are proposed and the accuracy are tested and compared with the semi-implicit schemes numerically.
Highlights
In this paper, we consider numerical schemes in the semi-discrete form to solve the Allen-Cahn equation φt − ∆φ + 1 ε2 f (φ) =∂φ (x, t) ∈ Ω × (0, T ), (1)∂n |∂Ω = 0, φ(t0) = φ0.where Ω ⊂ RN, N = 1, 2, 3 is a bounded domain with C1,1 boundary ∂Ω or a convex polygonal domain, n is the outward normal, f (φ) F ′(φ) and
The equation (1) was originally introduced by Allen-Cahn [1] to describe the motion of anti-phase boundaries in crystalline solids. φ represents the concentration of one of the two metallic components of the alloy and the parameter ε represents the interfacial length, which is extremely small compared to the characteristic dimensions on the laboratory scale
We derive a priori energy estimates which show that the first-order scheme is stable if its solution in some norm remains to be bounded as δt → 0
Summary
We consider numerical schemes in the semi-discrete form (in time) to solve the Allen-Cahn equation. Where Ω ⊂ RN , N = 1, 2, 3 is a bounded domain with C1,1 boundary ∂Ω or a convex polygonal domain, n is the outward normal, f (φ). The equation (1) was originally introduced by Allen-Cahn [1] to describe the motion of anti-phase boundaries in crystalline solids. The Allen-Cahn equation can be viewed as a gradient flow with Liapunov energy functional. The Allen-Cahn equation has been widely applied to many complicated moving interface problems, for example, vesicle membranes, the nucleation of solids and the mixture of two incompressible fluids, etc.
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