Local discrete-operator interpolation solution of the phase-field model

This paper reports continuing work on application of discrete-operator interpolation (DOI) in solving the one-dimensional phase-field model applied to melt-front tracking. DOI is a numerical technique for computing function values not computed at the original grid points of a finite-difference (or finite element) scheme so as to satisfy the discrete governing equations at the new points. The previous study showed that the DOI technique works quite well for the phase-field model problem. The shortcoming of earlier work was global (in space) application of DOI. Due to the fact that at any instant in time, the melt-front of the phase-field model exists within only a small region of space, it is more efficient to employ a local DOI technique. Local DOI interpolates the numerical solutions only in the melt-front region while a standard numerical method is applied in other regions. In this paper, we describe the phase-field model together with the details of the local DOI method and their numerical implementations. The results of the phase-field model are obtained using a Crank-Nicolson finite-difference scheme. The local DOI results are compared with direct numerical simulation results obtained on a very fine grid to demonstrate the advantages of this method.