A well posed and stable canonical evaporation model problem for phase-change in two-phase flows

We formulate a well posed interface formulation for canonical one-dimensional evaporation two-phase model problems (the Stefan and Sucking problems) commonly used to validate production codes. We focus on the interface between the vapor and the liquid and derive conditions leading to an energy bound and well-posedness. Next, by mimicking the continuous analysis, we discretize using high order accurate numerical methods on summation-by-parts form, impose the interface conditions weakly and prove energy stability.