Two-Level Finite Difference Methods for Simulating the High-Dimensional Lagging Models of Heat Conduction

ABSTRACT

Heat transfer at microscale plays an important role in microtechnology applications. First, this paper is concerned with the numerical solution of a two-dimensional (2D) heat transfer equation at microscale through a second-order alternating direction implicit (ADI) method. By the discrete energy method, it is shown that the ADI solution converges to the exact solution with a convergence order of in L²-norm. Second, the ADI method is generalized to numerically solve the three-dimensional (3D) problem, and provides numerical solutions of order two in both time and space. Finally, numerical results testify the temporal and spatial accuracy of the ADI method.

KEYWORDS:

• ADI method
• convergence
• finite difference method
• heat transfer equation at microscale

MATHEMATICS SUBJECT CLASSIFICATION:

• 65M06
• 65M12
• 65M15

Acknowledgments

Authors are deeply grateful to two anonymous referees, editor, and Principal Editor Prof. Zuhair Nashed for their insightful comments and helpful suggestions, which have greatly improved this article.