Multidimensional Stefan Problems

Abstract

An implicit finite difference method for the multidimensional Stefan problem is discussed. The classical problem with discontinuous enthalpy is replaced by an approximate Stefan problem with continuous piecewise linear enthalpy. An implicit time approximation reduces this formulation to a sequence of monotone elliptic problems which are solved by finite difference techniques. It is shown that the resulting nonlinear algebraic equations are solvable with a Gauss-Seidel method and that the discretized solution converges to the unique weak solution of the Stefan problem as the time and space mesh size approaches zero.

Finite volumes and nonlinear diffusion equations

“…Similar finite différence schemes were used by [2] and [12]. These authors show the convergence of the scheme.…”

The numerical solution of Stefan problems with front-tracking and smoothing methods

“…One particularly attractive method for the solution of It is known that the inner Gauss-Seidel iteration converges globally [27] , and it has been observed that the rate of convergence can be accelerated through point overrelaxation [18]. The choice of the relaxation parameter is critical for convergence.…”

“…Instead, we should like to concentrate on two specific and essentially unrelated numerical methods which have been described earlier [17], [18] and which have proved useful for a variety of Stefan type problems. The first of these methods so far has been used primarily for one dimensional free boundary and interface problems where the free surface varies smoothly with time.…”

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

“…The numerical analysis of the Stefan problem has been considered in several works. A finite difference method for the multi-dimensional Stefan problem is discussed in Meyer [32]. The author presents a convergent numerical scheme which is the implicit analogue of the method of Kamenomostskaja [27].…”