Numerical solution for the one-phase Stefan problem by piecewise-constant approximation of the interface
The classical one-phase one-dimensional Stefan problem is numerically solved on rectangles, \(R_ j\), of increasing size controlled by the Stefan condition. This approach is based on a scheme introduced by \textit{E. di Benedetto} and the second author [Rend. Semin. Mat. Univ. Padova 69, 109-134 (1983; Zbl 0476.35084)]. The practical implementation rests on the representation via thermal potentials of the solution \(u^ j(x,t)\) to the heat equation in \(R_ j\). The quantity \(u^ j_ x(x_ j,j\Delta t)\) which determines the \((j+1)\)-th rectangle is evaluated analytically by solving explicitly an integral equation. The solution in \(R_{j+1}\) is then obtained by numerically evaluating a further integral expression. The algorithm is tested by solving two problems whose solution is explicitly known. Convergence, stability and convergence rate as \(\Delta\) \(x\to 0\), \(\Delta\) \(t\to 0\) have been tested and plots are shown.
one-phase one-dimensional Stefan problem
scientific article; zbMATH DE number 4147451
scientific article