A conforming interface approach for phase transitions in rarefied gas dynamics

In rarefied gas dynamics, classical models like Fourier’s law cannot be used due to insufficient collisions and the lack of equilibrium. Extended gas dynamics models, such as moment approximations in kinetic theory, augment traditional continuum mechanics and include more variables and equations to describe the state of the gas, resulting in increased complexity of the equations. Similarly, the phase interface is subject to jump conditions that couple interface velocity, local equilibrium, and non-equilibrium variables in a possibly discontinuous way, triggering boundary layer effects that shape the bulk solution. Together with the nonlinear coupling of the evolving domain and the physical field equations, phase transitions in rarefied gas dynamics pose a significant challenge to numerical discretizations.

In this paper, we present a mesh-conforming interface method using standard finite elements and the level-set method with remeshing. At each time step, the domain is remeshed such that the computational mesh is conforming to the interface. This yields highly accurate and robust representations of the phase interface, allowing the interface conditions to be imposed directly and thus avoiding interpolation errors. Non-conforming methods, where an auxiliary function is used to represent the interface, lack those features.

The finite element solver and the meshing tool in our framework are exchangeable and adaptable. We provide an example implementation that is freely available and reusable in the reproducibility repository accompanying this paper. The framework is validated with simulations of the classical Stefan problem, and its flexibility is demonstrated on a model problem for rarefied gas heat conduction.