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

• Published in: Journal of computational physics Vol. 403; p. 109072
• Main Authors: Manzanero, Juan, Rubio, Gonzalo, Kopriva, David A., Ferrer, Esteban, Valero, Eusebio
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 15.02.2020; Elsevier Science Ltd
• Subjects: Approximation; Cahn–Hilliard; Computational physics; Dimensional stability; Discontinuous Galerkin; Fluxes; Galerkin method; High-order methods; Mathematical analysis; Summation–by–parts property; Discontinuous Galerkin; High-order methods; Summation–by–parts property; Cahn–Hilliard
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2019.109072

•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.