Fast and accurate numerical approaches for Stefan problems and crystal growth

New numerical approaches for moving boundary/interface applications tailored for Stefan problems and crystal growth simulation are proposed in this article. The focus is on the issues of accuracy and speed-up. A modified Crank-Nicolson method that is second-order accurate and stable is developed. The alternating directional implicit (ADI) method is also developed to speed up the simulation for a certain class of problems. The ADI method is shown to be asymptotically stable and at least first-order accurate. Numerical results, however, show that the ADI method actually provides second-order accuracy if the velocity can be calculated accurately. The level set method is used to update the moving interface so that the topological changes can be handled easily. Numerical experiments are compared to exact solutions and results in the literature.