Computation of Solid–Liquid Phase Fronts in the Sharp Interface Limit on Fixed Grids

Abstract

A finite-difference formulation is applied to track solid–liquid boundaries on a fixed underlying grid. The interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked. The imposition of boundary conditions exactly on a sharp interface that passes through the Cartesian grid is performed using simple stencil readjustments in the vicinity of the interface. Attention is paid to formulating difference schemes that are globally second-order accurate in x and t. Error analysis and grid refinement studies are performed for test problems involving the diffusion and convection–diffusion equations, and for stable solidification problems. Issues concerned with stability and change of phase of grid points in the evolution of solid–liquid phase fronts are also addressed. It is demonstrated that the field calculation is second-order accurate while the position of the phase front is calculated to first-order accuracy. Furthermore, the accuracy estimates hold for the cases where there is a property jump across the interface. Unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work. The results indicate the need to begin an effort to benchmark computations of instability phenomena.