On the multiple hyperbolic systems modelling phase transformation kinetics

Abstract

We discuss Cahn’s time cone method modelling phase transformation kinetics. The model equation by the time cone method is an integral equation in the space-time region. First, we reduce it to a system of hyperbolic equations, and in the case of odd spatial dimensions, the reduced system is a multiple hyperbolic equation. Next, we propose a numerical method for such a hyperbolic system. By means of alternating direction implicit methods, numerical simulations for practical forward problems are implemented with satisfactory accuracy and efficiency. In particular, in the three dimensional case, our numerical method on the basis of reduced multiple hyperbolic equation is fast.

Keywords:

• phase transformation
• Cahn’s time cone method
• multiple hyperbolic equation
• fast numerical method

AMS Subject Classifications:

• 74N05
• 35L30
• 65M06
• 65M99

Acknowledgments

The authors appreciate the invaluable discussions with Professors Jin Cheng, Wenbin Chen, Shuai Lu and Mr. Lingdi Wang (School of Mathematical Sciences, Fudan University). The authors are also grateful to the anonymous referee for useful comments. The first author is supported by the Leading Graduate Course for Frontiers of Mathematical Sciences and Physics of The University of Tokyo.