Second-Order Unconditionally Stable Direct Methods for Allen-Cahn and Conservative Allen-Cahn Equations on Surfaces

This paper describes temporally second-order unconditionally stable direct methods for Allen-Cahn and conservative Allen-Cahn equations on surfaces. The discretization is performed via a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation. We prove that the proposed schemes, which combine a linearly stabilized splitting scheme, are unconditionally energy-stable. The resulting system of discrete equations is linear and is simple to implement. Several numerical experiments are performed to demonstrate the performance of our proposed algorithm.