(Non-)convergence of solutions of the convective Allen–Cahn equation

We consider the sharp interface limit of a convective Allen–Cahn equation, which can be part of a Navier–Stokes/Allen–Cahn system, for different scalings of the mobility $$m_\varepsilon =m_0\varepsilon ^\theta $$ m ε = m 0 ε θ as $$\varepsilon \rightarrow 0$$ ε → 0 . In the case $$\theta >2$$ θ > 2 we show a (non-)convergence result in the sense that the concentrations converge to the solution of a transport equation, but they do not behave like a rescaled optimal profile in normal direction to the interface as in the case $$\theta =0$$ θ = 0 . Moreover, we show that an associated mean curvature functional does not converge to the corresponding functional for the sharp interface. Finally, we discuss the convergence in the case $$\theta =0,1$$ θ = 0 , 1 by the method of formally matched asymptotics.

Fast and Accurate Numerical Solution of Allen–Cahn Equation

Simulation speed depends on code structures. Hence, it is crucial how to build a fast algorithm. We solve the Allen–Cahn equation by an explicit finite difference method, so it requires grid calculations implemented by many for-loops in the simulation code. In terms of programming, many for-loops make the simulation speed slow. We propose a model architecture containing a pad and a convolution operation on the Allen–Cahn equation for fast computation while maintaining accuracy. Also, the GPU operation is used to boost up the speed more. In this way, the simulation of other differential equations can be improved. In this paper, various numerical simulations are conducted to confirm that the Allen–Cahn equation follows motion by mean curvature and phase separation in two-dimensional and three-dimensional spaces. Finally, we demonstrate that our algorithm is much faster than an unoptimized code and the CPU operation.