A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation

We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard equation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The latter permits us to show that the discrete free-energy is bounded, and as a result, the scheme is provably stable. The scheme and the stability proof are presented for general curvilinear three-dimensional hexahedral meshes. We use the Bassi-Rebay 1 (BR1) scheme to compute interface fluxes, and an IMplicit-EXplicit (IMEX) scheme to integrate in time. Lastly, we test the theoretical findings numerically and present examples for two and three-dimensional problems.