Open Access

ARTICLE

Past Cone Dynamics and Backward Group Preserving Schemes for Backward Heat Conduction Problems

by C.-S. Liu¹, C.-W. Chang², J.-R. Chang²

1 Corresponding author. E-mail: csliu@mail.ntou.edu.tw, Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung, Taiwan
2 Department of Systems Engineering and Naval Architecture, Taiwan Ocean University, Keelung, Taiwan

Computer Modeling in Engineering & Sciences 2006, 12(1), 67-82. https://doi.org/10.3970/cmes.2006.012.067

Abstract

In this paper we are concerned with the backward problems governed by differential equations. It is a first time that we can construct a backward time dynamics on the past cone, such that an augmented dynamical system of the Lie type X^˙ = B(X,t)X with t ∈ R⁻, X ∈ Mⁿ⁺¹ lying on the past cone and B ∈ so(n,1), was derived for the backward differential equations system x^· =f(x,t), t ∈ R⁻, x ∈ Rⁿ. These two differential equations systems are mathematically equivalent. Then we apply the backward group preserving scheme (BGPS), which is an explicit single-step algorithm formulated by an exponential mapping to preserve the group preperties of SO_o(n,1), on the backward heat conduction problem (BHCP). It can retrieve all the initial data with high order accuracy. Several numerical examples of the BHCP were work out, and we show that the BGPS is applicable to the BHCP, even those of strongly ill-posed ones. Under the noisy final data the BGPS is also robust to against the disturbance. The one-step BGPS effectively reconstructs the initial data from a given final data, with a suitable grid length resulting into a high accuracy never seen before. The results are very significant in the computations of BHCP.

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Keywords

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