High Accuracy Stable Numerical Solution of
1D Microscale Heat Transport Equation

Jun Zhang
Department of Computer Science
University of Kentucky
773 Anderson Hall
Lexington, KY 40506-0046, USA

Jennifer J. Zhao
Department of Mathematics and Statistics
University of Michigan at Dearborn
Dearborn, MI 48128-1491, USA

Abstract

We investigate the use of a fourth order compact finite difference scheme for solving an one dimensional heat transport equation at the microscale. The fourth order compact scheme is used with a Crank-Nicholson type integrator by introducing an intermediate function for the heat transport equation. The new scheme is proved to be unconditionally stable with respect to initial values. Numerical experiments are conducted to compare the new scheme with the existing scheme based on second order spatial discretization. It is shown that the new scheme is computationally more efficient and more accurate than the second order scheme.


Key words: Heat transport equation, finite difference, fourth order compact scheme, Crank-Nicholson integrator.

Mathematics Subject Classification: 65M06, 65N12.


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This paper has been published in Communications in Numerical Methods in Engineering, Vol. 17, No. 11, pp. 821-832 (2001). Technical Report 297-00, Department of Computer Science, University of Kentucky, Lexington, KY, 2000. This research was supported in part by the U.S. National Science Foundation under the grant CCR-9902022, and in part by the University of Kentucky Center for Computational Sciences.