Long Time Numerical Simulations for Phase-Field Problems Using $p$-Adaptive Spectral Deferred Correction Methods

Abstract

A high-order and energy stable scheme is developed to simulate phase-field models by combining the semi-implicit spectral deferred correction (SDC) method and the energy stable convex splitting technique. The convex splitting scheme we use here is a linear unconditionally stable method but is only of first-order accuracy, so the SDC method can be used to iteratively improve the rate of convergence. However, it is found that the accuracy improvement may affect the overall energy stability which is intrinsic to the phase-field models. To compromise the accuracy and stability, a local p-adaptive strategy is proposed to enhance the accuracy by sacrificing some local energy stability in an acceptable level. The proposed strategy is found very useful for producing accurate numerical solutions at small time (dynamics) as well as long time (steady state) with reasonably large time stepsizes. Numerical experiments are carried out to demonstrate the high effectiveness of the proposed numerical strategy.

Unconditionally maximum principle preserving finite element schemes for the surface Allen–Cahn type equations

“…In general, the parameter S is set to be S ≥ 2 [4,14,21]. In general, the parameter S is set to be S ≥ 2 [4,14,21].…”

“…In terms of this topic, the stabilized semi-implicit finite difference scheme is developed and has been proved to be maximum principle preserving [13,14,21]. However, the stronger stability under the infinity norm has a great effect on avoiding numerical oscillations.…”

Investigations on several numerical methods for the non-local Allen–Cahn equation

“…It has been widely applied to various problems, such as crystal growth [2], image analysis [3,4] and phase separation [5], etc. Compared to large amount of studies for the classical Allen-Cahn equation [6][7][8][9], there are few numerical results on non-local Allen-Cahn equation. In 1992, Rubinstein and Sternberg [10] first proposed non-local Allen-Cahn equation, which is related to the mean-curvature flow with the constraint of constant volume enclosed by the evolving curve.…”

“…Meanwhile, we assume the initial data ju 0 j 6 1, which follows from the maximum principe that juj 6 1 in [11]. The term bðu; tÞ can be understood as a Lagrange multiplier for the mass constraint Both formulations have been widely used in [9,[13][14][15][16]. Notice that the mass conservation Eq.…”

“…Future research will concern the extension of the Fourier spectral operator splitting method to temporal adaptive algorithm [9,23,24], as well as the simulation of the 3D Cahn-Hilliard equation [9,25].…”