A rapidly converging phase field model

Abstract

Abstract. We propose a phase field model that approximates its limiting sharp interface model (free boundary problem) up to second order in interface thickness. A broad range of double-well potentials can be utilized so long as the dynamical coefficient in the phase equation is adjusted appropriately. This model thereby assures that computation with particular value of interface thickness ε, will differ at most by O(ε 2 ) from the limiting sharp interface problem. As an illustration, the speed of a traveling wave of the phase field model is asymptotically expanded to demonstrate that it differs from the speed of the traveling wave of the limit problem by O(ε 2 ). 1.Introduction. Interface problems arising from solidification have been studied extensively in mathematics and physics for more than a century [29]. The mathematical study began with Lamè and Clapeyron [23] in 1831 who modeled the freezing of the ground using the heat equation, latent heat across the interface, and the condition that the temperature at the interface remains at the equilibrium freezing temperature. Reformulated in 1889, this problem became known as the classical Stefan model [31], and can be stated as follows. Determine the temperature, T (x, t), and the interface, Γ(t), satisfying the system of equations,

Phase Field Models Versus Parametric Front Tracking Methods: Are They Accurate and Computationally Efficient?

“…in place of (3.9a,b), the authors in [32] also show rigorously that the full phase field converges to second order. More precisely, in this case the first order correction to the phase field function ϕ is zero.…”

“…In [32], for the special case (s) = 1 2 , the above second order approximation results are shown rigorously. In particular, on letting K = a = 1, and on recalling that in their notation G(s) = c Ψ P(s), it holds that the expression in [32, Eq.…”

“…In this subsection we recall a variant of the phase field model (3.5a-d), (3.6a-c) which in the isotropic setting (2.2), (2.4) yields a second order convergence in ε to the sharp interface limit; see e.g. [52,3,45,32,28] for details. In particular, it needs to be assumed that the shape function , recall (3.7), satisfies (3.9a,b).…”

“…Their approach made it possible to do efficient computations with a smaller capillary length to interface thickness ratio, and to study the physically relevant case of small or zero kinetic coefficients. Later the findings of Karma and Rappel were reinterpreted as second order convergence with respect to the interfacial thickness, see [3,45,32].…”

Improved phase-field models of melting and dissolution in multi-component flows

“…While variational derivations are useful, it is asymptotic analysis, tested with numerical experiments, that best determines the accuracy of phase-field models. Proving these equations converge to the moving boundary formulation at Ofalse(ε2false) in general geometries is nontrivial, but builds on second-order models of each individual boundary condition; a phase-field model which optimizes the mobility term for zero interface kinetics [59], a concentration equation similar to [31] and the diffuse domain method for Robin boundary conditions [60], and the smooth volume penalty method (which gives β = 1.51044385) [61].…”