An inverse problem for the one-phase Stefan problem with varying melting temperature

The present article is dedicated to the forward and backward solution of a transient one-phase Stefan problem. In the forward problem, we compute the evolution of the initial domain for a Stefan problem where the melting temperature varies over time. This occurs in practice, for example, when the pressure in the external space changes in time. In the corresponding backward problem, we then reconstruct the time-dependent melting temperature from the knowledge of the evolving geometry. We develop respective numerical algorithms using a moving mesh finite element method and provide numerical simulations.

💡 Research Summary

The paper addresses both the forward and inverse formulations of a transient one‑phase Stefan problem in which the melting temperature at the phase interface is not constant but varies with time, a situation that can arise, for example, when external pressure changes. In the forward problem the authors first rewrite the classical Stefan model by introducing the shifted temperature variable (v = u - u_m(t)). This transformation converts the heat equation into a non‑homogeneous diffusion equation (\partial_t v - \Delta v = -\dot u_m(t)) while preserving the Stefan condition (\langle V,n\rangle = -\partial_n v) on the moving boundary.

Time is discretised into steps (t_k = k\Delta t). For each step a fixed spatial domain (\Omega_k) is considered; the domain is assumed to be star‑shaped and its boundary (\Gamma_k) is represented by a truncated Fourier series with (2M+1) coefficients. The Fourier representation enables the use of the Fast Fourier Transform to update the boundary efficiently. The heat equation on (\Omega_k) is solved with a Crank–Nicolson finite‑element scheme, leading to a linear system ((M_k + \frac{\Delta t}{2}A_k)v^{k+1} = (M_k - \frac{\Delta t}{2}A_k)v^{k} - \Delta t,\alpha_k f_k), where (M_k) and (A_k) are the mass and stiffness matrices, (\alpha_k) approximates (\dot u_m) on the interval, and (f_k) is the unit source term.

After obtaining (v^{k+1}), the Stefan condition provides the normal velocity of the interface, which is used to move each boundary point radially according to (x^{k+1}=x^{k}-\Delta t,\partial_n v^{k+1}\langle n,b_x\rangle b_x). The new boundary points are projected back onto the Fourier basis (least‑squares fit) to obtain the updated coefficient vector for the next time step. Meshes on the deformed domains are generated by mapping a reference mesh on the unit disc through the affine mapping defined by the Fourier coefficients, guaranteeing mesh conformity across time steps.

The inverse problem is formulated as follows: given a time series of observed free‑boundary positions (\Gamma(t)), recover the unknown melting temperature function (u_m(t)). The authors analyse identifiability, showing that if (u_m(t)) is sufficiently smooth and the boundary data are noise‑free (or only mildly noisy), the reconstruction is unique. Numerically, synthetic data are generated by prescribing a known (u_m(t)) (e.g., a sinusoidal function) and running the forward solver to obtain (\Gamma(t)). An iterative inversion, essentially a gradient‑based optimisation that minimizes the mismatch between simulated and observed boundaries while updating (u_m(t)), successfully recovers the original temperature profile. Reconstruction errors decrease with finer time steps and higher Fourier truncation order, at the cost of increased computational effort.

Robustness against measurement noise is demonstrated by adding Gaussian perturbations (≈1 % standard deviation) to the boundary data and employing Tikhonov regularisation together with temporal smoothing. The recovered (u_m(t)) remains close to the true profile, confirming the practical viability of the approach for experimental settings where only the interface motion can be measured (e.g., pressure‑controlled methane hydrate growth).

The paper’s contributions are threefold: (1) a unified forward–inverse numerical framework for Stefan problems with time‑dependent melting temperature, (2) an efficient moving‑mesh finite‑element method that leverages Fourier boundary parametrisation and FFT, and (3) a demonstration that the moving interface alone contains sufficient information to reconstruct the temperature function, opening the way to parameter identification in pressure‑sensitive phase‑change systems. The authors suggest extensions to three‑dimensional geometries, coupling with physical pressure‑temperature laws, and parallel implementation as future work.