Solving two typical inverse Stefan problems by using the Lie-group shooting method

We consider two typical inverse Stefan problems: one is computing a heat flux boundary condition when the moving boundary xi(t) is given, and another is recovering an unknown moving boundary xi(t), by knowing an extra Dirichlet boundary condition on the accessible boundary. Through a domain embedding method, we can transform the inverse problem into a parameter identification problem of an advection-diffusion partial differential equation, where xi(t) and xi(t) play the role of unknown parameters for the second inverse problem. The xi(t) appeared in the governing equation makes the identification of xi(t) rather difficult. However, upon using the Lie-group shooting method (LGSM) we can derive a simple system of algebraic equations to iteratively calculate xi(t) and then xi(t) at some discretized times. It is demonstrated through numerical examples that the LGSM is accurate and stable, although under a large measurement noise. (C) 2011 Elsevier Ltd. All rights reserved.