A free–energy stable nodal discontinuous Galerkin approximation with summation–by–parts property for the Cahn–Hilliard equation - Manzanero, Juan; Rubio, Gonzalo; Kopriva, David A.; Ferrer, Esteban; Valero, Eusebio (Journal of computational physics, 2020)
•Discontinuous Galerkin approximation for the Cahn—Hilliard eqn with SBP—SAT property.•The scheme uses Gauss—Lobatto points, and BR1 interface fluxes for the approximation.•Semi—discrete in space free—energy stability proof.•Fully—discrete stability proof using a first order implicit—explicit Euler scheme.•Scheme and stability proof hold for general unstructured 3D curvilinear hex—meshes. 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 a first order IMplicit–EXplicit (IMEX) scheme to integrate in time. We provide a semi–discrete stability study, and a fully–discrete proof subject to a positivity condition on the solution. Lastly, we test the theoretical findings using numerical cases that include two and three–dimensional problems.