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

Abstract

In this paper we are concerned with the backward problems governed by differential equations. It is afirst timethatwe can constructabackward timedy- namics on the past cone, such that an augmented dynam- ical system of the Lie type ˙ X = B(X,t)X with t ∈ R − , X ∈ M n+1 lying on the past cone and B ∈ so(n,1) ,w as derived for the backward differential equations system ˙ x = f(x,t), t ∈ R − ,x ∈ R n . These two differential equa- tions 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 SOo(n,1), on the backward heat conduction problem (BHCP). It can retrieve all the initialdata withhigh order accuracy. Several numerical examples of the BHCP were work out,and weshow that theBGPS is applicableto 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 be- fore. The results are very significant in the computations of BHCP. keyword: Past cone dynamics, Backward group pre- serving scheme, Backward heat conduction problem, Strongly ill-posed problem.