A Backward Group Preserving Scheme for Multi-Dimensional Backward Heat Conduction Problems

Abstract

In this article, we propose a backward group preserving scheme (BGPS) to tackle the multi-dimensional backward heat conduction problem (BHCP). The BHCP is well-known as severely ill-posed because the solution does not continu- ously depend on the given data. When eight numerical examples (including non- linear and nonhomogeneous BHCP, and Neumann and Robin conditions of homo- geneous BHCP) are examined, we find that the BGPS is applicable to the multi- dimensional BHCP. Even with noisy final data, the BGPS is also robust against disturbance. The one-step BGPS effectively reconstructs the initial data from the given final data, which with a suitable grid length produces a highly accurate so- lution never seen before. The results are very important in the computations of multi-dimensional BHCP.