Numerical error analysis for an energy-stable HDG method for the Allen–Cahn equation

This paper presents an energy-stable hybridizable interior penalty discontinuous Galerkin method for the Allen–Cahn equation. To obtain an unconditionally energy stable scheme, the energy potential is split into a sum of a convex and concave function. Energy stability for the proposed scheme is proven to hold for arbitrary time. Existence and uniqueness for the scheme is also established. Under standard assumptions on the energy potential (Lipschitz continuity), we demonstrate rigorously that the method converges optimally for symmetric schemes, and suboptimally for nonsymmetric schemes. Several examples are provided which numerically verify and validate the method.