Geometric Conservation Law for Finite Volume Discretization of the Stefan Problem on Boundary-Fitted Grids

A conservative finite volume scheme for heat transfer problem in a two-dimensionaldomain with moving boundaries is presented. The two-phase Stefan problem is considered as anexample. The front-fixing technique is applied to track the moving interface between the solid andthe liquid. The time-varying physical domain is mapped to a fixed computational space withregular boundaries. Finite volume approximation to governing equations is constructed in thecomputational domain on a fixed rectangular grid. The geometric conservation law is incorporatedin the numerical scheme. The Jacobian and the grid velocities of the control volume are evaluatedto satisfy the discrete form of the Jacobian transport equation. This procedure guarantees theenforcing of the space conservation law in the physical domain. The numerical scheme inherits thebasic properties of the original differential problem.