Group preserving scheme for backward heat conduction problems

In this paper we numerically integrate the backward heat conduction equation ∂u/∂t=ν▵u, in which the Dirichlet boundary conditions are specified at the boundary of a certain spatial domain and a final data is specified at time T>0. In order to treat this ill-posed problem we first convert it through the transformation s=T−t to an unstable initial-boundary-value problem: ∂u/∂s=−ν▵u together with the same boundary conditions and the same data at s=0. Then, we consider the contraction map of u to v=exp[−as]u by a suitable contraction factor a>0, which is analyzed by considering the stability of the semi-discretization numerical schemes. The resulting ordinary differential equations at the interior grid points are then numerically integrated by the group preserving scheme, proposed by Liu [Int. J. Non-Linear Mech. 36 (2001) 1047], and the stable range of the index r=νΔt/(Δx)2 is derived. Numerical tests for both forward and backward heat conduction problems are performed to confirm the effectiveness of the new numerical methods.