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 ξ(t) is given, and another is recovering an unknown moving boundary ξ(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 ξ(t) and image play the role of unknown parameters for the second inverse problem. The image appeared in the governing equation makes the identification of ξ(t) rather difficult. However, upon using the Lie-group shooting method (LGSM) we can derive a simple system of algebraic equations to iteratively calculate image and then ξ(t) at some discretized times. It is demonstrated through numerical examples that the LGSM is accurate and stable, although under a large measurement noise.