A modified Lie-group shooting method for multi-dimensional backward heat conduction problems under long time span

This paper proposes a modified Lie-group shooting method to solve multi-dimensional backward heat conduction problems under long time spans. The backward heat conduction problem is renowned for being ill posed because the solutions are generally unstable and highly dependent on the given data. For dealing with those problems, the Lie-group shooting method is one of the most powerful tools to find the unknown initial condition for the backward heat conduction problems in the time domain. In previous studies, the Lie-group shooting method uses the time and spatial semi-discretization technique to change the integration direction of numerical schemes and then increase the time span. However, the conversional Lie-group shooting method cannot get to the core of divergence problems for the backward heat conduction problems, especially the increased computational time. The main reason is that a real single-parameter Lie-group element occurs at zero, and a generalized midpoint Lie-group element is not equivalent to the single-parameter Lie-group element in the Minkowski space. Hence, to overcome the above problems, the relationship of the initial condition, the final condition and a real single-parameter r is assessed. According to the constraint condition of the initial and final condition, a real single-parameter r depends on the time span to maintain the numerical convergence. Again, in order to preserve the same Lie-group property in the time direction, the high-order Lie-group scheme based on the generalized linear group in Euclidean space is introduced, which concurrently satisfies the constraint of the cone structure, the Lie-group and the Lie algebra at each time step. The accuracy and efficiency are validated, even under noisy measurement data, by comparing the estimation results with existing literature.