Fast Numerical Simulation of Allen-Cahn Equation

TL;DR

This paper introduces a GPU-accelerated convolutional approach to efficiently simulate the Allen-Cahn equation, significantly improving speed over traditional CPU-based finite difference methods.

Contribution

It proposes a novel convolutional model architecture with padding for the Allen-Cahn equation, leveraging GPU operations to enhance simulation speed.

Findings

GPU-based convolutional method outperforms CPU implementation

Simulation confirms Allen-Cahn equation models mean curvature motion

Algorithm achieves faster results than unoptimized code

Abstract

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. To solve the problem, we propose a model architecture containing a pad and a convolution operation for the Allen-Cahn equation. 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.