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

## TL;DR

This paper introduces a stable nodal discontinuous Galerkin scheme for the Cahn-Hilliard equation that guarantees free-energy boundedness and stability on complex 3D meshes, validated through numerical tests.

## Contribution

The paper develops a novel DG scheme satisfying the SBP-SAT property for stability, applicable to general curvilinear meshes, and demonstrates its effectiveness through numerical experiments.

## Key findings

- The scheme is proven to be free-energy stable.
- Numerical tests confirm the theoretical stability and accuracy.
- Applicable to complex 3D geometries with curvilinear meshes.

## Abstract

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.