A Fictitious Time Integration Method for Multi-Dimensional Backward Heat Conduction Problems

摘要

In this article, we propose a new numerical approach for solving these multi-dimensional nonlinear and nonhomogeneous backward heat conduction problems (BHCPs). A fictitious time T is employed to transform the dependent variable u(x, y, z, t) into a new one by (1+τ)(x, y, z, t)=: v(x, y, z, t, τ), such that the original nonlinear and nonhomogeneous heat conduction equation is written as a new parabolic type partial differential equation in the space of (x, y, z, t, τ). In addition, a fictitious viscous damping coefficient can be used to strengthen the stability of numerical integration of the discretized equations by utilizing a group preserving scheme. Six numerical experiments illustrate that the present algorism can be employed to recover the initial data very well. Even under the very large noisy final data, the fictitious time integration method is also robust against noise.