Dissipation-preserving discretization of the Cahn--Hilliard equation with dynamic boundary conditions

TL;DR

This paper introduces new time-stepping schemes for the Cahn--Hilliard equation with dynamic boundary conditions that preserve mass and energy dissipation, allowing flexible spatial discretizations and computational efficiency.

Contribution

It proposes first and second order schemes that are mass-conservative and energy-dissipative, enabling boundary refinements without interior mesh changes.

Findings

Schemes are mass-conservative and energy-dissipative.

Numerical experiments demonstrate computational gains.

Flexible boundary discretizations improve efficiency.

Abstract

This paper deals with time stepping schemes for the Cahn--Hilliard equation with three different types of dynamic boundary conditions. The proposed schemes of first and second order are mass-conservative and energy-dissipative and -- as they are based on a formulation as a coupled system of partial differential equations -- allow different spatial discretizations in the bulk and on the boundary. The latter enables refinements on the boundary without an adaptation of the mesh in the interior of the domain. The resulting computational gain is illustrated in numerical experiments.