A highly accurate backward-forward algorithm for multi-dimensional backward heat conduction problems in fictitious time domains

This paper proposes highly accurate one-step backward-forward algorithms for solving multi-dimensional backward heat conduction problems (BHCPs). The BHCP is renowned for being ill-posed because the solutions are generally unstable and highly dependent on the given data. In this paper, the present algorithm combines algebraic equations with a high-order Lie-group scheme to construct one-step algorithms called the backward fictitious integrate method (BFTIM) and the forward fictitious integrate method (FFTIM). First, the original parabolic equation is transformed into a new parabolic equation of an evolution type by introducing a fictitious time variable. Then, the numerical integration of the discretized algebraic equations must satisfy the constraints of the cone structure, Lie-group and Lie algebra at each fictitious time step. Finally, the algorithms with the minimum fictitious time steps along the manifold of the Lie-group scheme approach the true solution with one step when given an initial guess. In addition, this paper provides a strategy to determine the initial guess, which is the reciprocal relationship of the initial condition (IC) and the final condition (FC). More importantly, the IC and FC can be recovered by the BFTIM and FFTIM according to the relation between the IC and FC, even under large noisy measurement data. Five numerical examples of the BHCP are tested and numerical results demonstrate that the present schemes are more effective and stable. In general, the numerical implementations of the BFTIM and FFTIM are simple and have one-step convergence speeds.