Jimmy Kornelije Gunnarsson

Jimmy Kornelije Gunnarsson, Robert Klöfkorn

The Cahn-Hilliard-Navier-Stokes (CHNS) system utilizes a diffusive phase-field for interface tracking of multi-phase fluid flows. Recently structure preserving methods for CHNS have moved into focus to construct numerical schemes that, for example, are mass conservative or obey initial bounds of the phase-field variable. In this work decoupled implicit-explicit formulations based on the Discontinuous Galerkin (DG) methodology are considered and compared to existing schemes from the literature. For the fluid flow a standard continuous Galerkin approach is applied. An adaptive conforming grid is utilized to further draw computational focus on the interface regions, while coarser meshes are utilized around pure phases. All presented methods are compared against each other in terms of bound preservation, mass conservation, and energy dissipation for different examples found in the literature, including a classical rising droplet problem.

Jimmy Kornelije Gunnarsson, Robert Klöfkorn

We propose a structure-preserving discontinuous Galerkin scheme for the Cahn--Hilliard equations with degenerate mobility based on the Symmetric Weighted Interior Penalty formulation. By evaluating the mobility at cell averages rather than as a piecewise polynomial, the proposed scheme preserves strict degeneracy and yields a coercivity constant that is independent of the mobility, removing the need for regularisation. Moreover, we establish existence of discrete solutions even with degeneracy via a Leray--Schauder fixed-point argument, and show that the scheme satisfies a provable discrete maximum principle at arbitrary polynomial order $p$ when combined with the Zhang--Shu scaling limiter for $p > 0$ and from the scheme alone for $p = 0$. Mass conservation and energy dissipation are established for the unlimited scheme; for the limited variant, we discuss observed energy dissipation for $p \geq 1$ and potential theoretical solutions. Numerical experiments confirm optimal convergence rates of order $p+1$ in $L^2$ and validate structure-preserving properties with numerical results.