Second order front tracking algorithm for Stefan problem on a regular grid

TL;DR

This paper introduces a second order accurate front tracking algorithm for the Stefan problem on regular grids, enabling precise and mass-conserving simulations of solidification in multiple dimensions with modest computational resources.

Contribution

A novel second order front tracking method for the Stefan problem that conserves mass and improves interface motion accuracy on regular grids.

Findings

Achieves 5.5% accuracy with 5 grid points in 1D

Achieves 1.5% accuracy with 10 grid points in 1D

Applicable to 2D and 3D simulations with ease

Abstract

A brief review of the Stefan problem of solidification from a mixture, and its main numerical solution methods is given. Simulation of this problem in 2D or 3D is most practically done on a regular grid, where a sharp solid-liquid interface moves relative to the grid. For this problem, a new simulation method is developed that manifestly conserves mass, and that simulates the motion of the interface to second order in the grid size. When applied to an isothermal simulation of solidification from solution in 1D at 50% supersaturation for only 5 grid points, the motion of the interface is accurate to 5.5%; and for 10 points the result is accurate to 1.5%. The method should be applicable to 2D or 3D with relative ease. This opens the door to large scale simulations with modest computer power.