Error estimates for the explicit finite difference method solving the Stefan problem (Q1119005)

The explicit finite difference method for solving the one-dimensional two-phase Stefan problem, going back to \textit{S. L. Kamenostskaja} [Mat. Sbornik 53(95), 489-514 (1961; Zbl 0102.093)]is analyzed. Under assumptions on the data which assure a regularity \(u\in W_ 2^{1,1}(\Omega_ T)\cap L_{\infty}(0,T;W^ 1_ 2(\Omega))\) and \(s\in W^ 1_ 4(0,T)\) of the temperature u and the moving boundary s \(L_ 2\)-convergence rates O(h) for the approximations of u and s are derived, where h is the local discretization step. Additionally, a case of much weaker regularity is considered.