Fourth-Order Compact Difference Scheme for the Backward Heat Conduction Problem

Abstract

In this paper, the backward heat conduction problem is solved numerically using the fourth-order compact difference scheme. For regularization of this ill-posed problem, the quasi-reversibility technique is used. Compact finite difference approximation of the second derivative has been compared with the fourth-order standard finite difference approximation via Fourier analysis. The comparison shows that the resolution ability of the compact difference approximation is better than the standard finite difference approximation. The discrete dispersion relation for the compact difference scheme and finite difference scheme is obtained. Two test problems are considered for the numerical experiments. The CPU time taken by a fourth-order compact difference scheme is obtained and compared with the fourth-order standard finite difference scheme which shows that the fourth-order compact difference scheme takes lesser CPU time.

Keywords:

• Backward heat conduction problem
• compact difference scheme
• ill-posed problems
• dispersion relation
• quasi-reversibility

Acknowledgement

The second author also acknowledge the support provided by the Departmentof Science and Technology, India, under the grant number MTR/2018/000371.