A multigrid solver for the Allen-Cahn equation on a virtual cubic surface

초록

We present an unconditionally stable hybrid finite difference scheme with a multigrid solver for the nonlinear Allen-Cahn (AC) model on a virtual cubic surface. Previous fully explicit operator splitting methods suffer from strict time step restrictions that limit computational efficiency. To overcome this, the diffusion term is discretized implicitly and solved using a V-cycle multigrid algorithm, while the nonlinear term is updated analytically in closed form. The proposed scheme incorporates boundary conditions reflecting the virtual cubic surface geometry and achieves second-order spatial accuracy and first-order temporal accuracy. Numerical tests confirm that the proposed algorithm preserves the discrete maximum principle and satisfies the energy dissipation property within a multigrid tolerance, even for large time steps beyond the explicit stability limit. Computational experiments include convergence analysis, validation of energy and maximum principle preservation, and simulations of motion by mean curvature and traveling wave propagation on the cubic surface. Computational results demonstrate that the scheme accurately captures interface dynamics, including curvature-driven motion and complex shape evolution, and maintains both stability and efficiency. The use of the multigrid method ensures fast convergence for the diffusion step, and the approach is well suited for large-scale simulations of phase-field models on curved or piecewise-planar surfaces. This work provides a practical and robust computational methodology for solving the nonlinear AC equation on the virtual surfaces of a rectangular cuboid, and it enables efficient simulation of interfacial phenomena with reduced computational cost and without restrictive time step constraints.

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