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

Abstract

Based on the idea of the Lie-group shooting method and the backward group preserving scheme, a novel backward-forward algorithm is developed to solve non-linear, nonhomogeneous, multi-dimensional backward heat conduction problems under long time spans. For a nonhomogeneous, multi-dimensional, backward heat conduction problem, it is very difficult to integrate towards the time direction, even when using a high-order numerical scheme. To avoid time integration and increase the computational efficiency, a novel backward-forward Lie-group scheme is proposed. According to the quadratic equation of the Lie-group shooting method, a solution is applied to obtain the initial condition and to examine the final condition. Using the reciprocal relationship between the solutions of forward and backward schemes, the proposed algorithm can avoid the time integration of the numerical scheme and numerical divergence. To illustrate the effectiveness and accuracy of the proposed algorithm, several benchmarks in multi-dimensions are tested. The numerical results indicate that the proposed algorithm can achieve an efficient and accurate solution, even with noisy measurement data by comparing the estimation results with the existing literatures.