TITLE:
Crank-Nicolson Finite Difference Scheme for the 3D Allen-Cahn Equation with PINN Comparison
AUTHORS:
Bo Liu, Lijuan Wang
KEYWORDS:
Allen-Cahn Equation, Crank-Nicolson Scheme, Maximum Principle, Physics-Informed Neural Network (PINN)
JOURNAL NAME:
Applied Mathematics,
Vol.16 No.10,
October
31,
2025
ABSTRACT: In this paper, we derive a second-order Crank-Nicolson finite-difference scheme for the three-dimensional Allen-Cahn equation and present an estimate of the scheme’s truncation error. We prove the existence of the numerical solution and establish stepwise uniqueness, from which the global uniqueness follows given the initial value u 0. We also establish conditional convergence in the L∞ -norm, and demonstrate a discrete maximum principle for the scheme. Numerical Examples 1 and 2 validate the discrete maximum principle, while Numerical Example 3 employs a physics-informed neural network (PINN), using the Crank-Nicolson solution as the reference, to compute the errors between the two approaches in the L2 and L∞ norms. Owing to the high dimensionality of the problem, visualization is performed via isosurface plots on identical time slices; these plots show strong agreement between the Crank-Nicolson and PINN solutions. The numerical results collectively confirm the accuracy and effectiveness of the proposed scheme.