Past Cone Dynamics and Backward Group Preserving Schemes for Backward Heat Conduction Problems
C.-S. Liu; C.-W. Chang; J.-R. Chang

doi:10.3970/cmes.2006.012.067
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 12, No. 1, pp. 67-82, 2006
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Keywords Past cone dynamics, Backward group preserving scheme, Backward heat conduction problem, Strongly ill-posed problem.
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$\mathaccentV {dot}05F{\bf X}={\bf B}({\bf X},t){\bf X}$ with$t \in {\@mathbb R}^-$,${\bf X} \in {\@mathbb M}^{n+1}$ lying on the past cone and${\bf B} \in so(n,1)$, was derived for the backward differential equations system$\mathaccentV {dot}05F{\bf x} = {\bf f}({\bf x},t)$,$t \in {\@mathbb R}^{-}, {\bf x} \in {\@mathbb R}^n$. 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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