On a boundary updating method for the scalar Stefan problem

Abstract In the present work, we propose a general-purpose method for solving the scalar Stefan problem, leveraging a modified version of a well-known boundary updating approach, commonly used in standard existence proofs. Specifically, we consider a Stefan problem with a prescribed positive heat flux at the initial time. Our approach introduces an operator that modifies the phase boundary, such that the fixed points of this operator correspond to solutions of the Stefan problem. The operator is constructed as an integral form of the Stefan condition, and it is shown to be continuous in an appropriate norm. As a result, the existence of a fixed point follows. Moreover, for sufficiently small time horizons, the operator becomes a contraction, enabling a potential method for numerically identifying the boundary. The proposed modification facilitates the numerical solution of a broad class of free boundary problems. We provide a rigorous theoretical justification of the method, along with several computational results that numerically validate its effectiveness. Finally, we present a spectral analysis of the matrix sequence that arises in the discretization of the Stefan problem. Using tools from Generalized Locally Toeplitz (GLT) theory, we characterize the eigenvalue and singular value distributions of the relevant matrix sequence. We identify its spectral symbol and provide a GLT momentary symbol expansion. This analysis enables the design of efficient solvers for the resulting linear systems, and numerical experiments illustrate the method’s performance.