A Stable Time Discretization of the Stefan Problem with Surface Tension

Abstract

Abstract. We present a time discretization for the single phase Stefan problem with GibbsThomson law. The method resembles an operator splitting scheme with an evolution step for the temperature distribution and a transport step for the dynamics of the free boundary. The evolution step involves only the solution of a linear equation that is posed on the old domain. We prove that the proposed scheme is stable in function spaces of high regularity. In the limit ∆t → 0 we find strong solutions of the continuous problem. This proves consistency of the scheme, and additionally it yields a new short-time existence result for the continuous problem. 1. Introduction. The Stefan problem is a set of equations that describe the melting of ice or the growth of ice cristals. At time t the ice (or the water) occupies a region Ω t , and the second phase occupies the complement of Ω t . The position of the interface ∂Ω t is not known a priori but must be determined together with the temperature distribution Θ(t). Several sets of evolution equations can be found in the extensive literature (see [11] for equations and further references). Commonly used is the heat equation (2.1) in the domain Ω t (in the two phase problem another heat equation is posed in the complement of Ω t ). The latent heat relates the normal heat flux on the free boundary (or its jump across the boundary) with the speed of the free boundary as in (2.2). In order to determine the evolution we need one more boundary condition. Various possibilities are studied for that: (a) fixed temperature Θ = 0, (b) the Gibbs-Thomson relation Θ ∼ κ with κ being the mean curvature of the boundary, and (c) kinetic undercooling: temperature plus a multiple of the speed is proportional to the mean curvature. In the paper at hand we are interested in case (b), the Gibbs-Thomson relation (2.3).The aim of this paper is to introduce a stable time discretization of the twodimensional free boundary value problem. We consider the single phase problem for simplicity; the two phase problem can be treated with the same method. Since the domain changes with time, it is not clear what equations we should pose at every time step, how to define a new domain, and how to define a temperature distribution on the new domain. Thinking of the numerical use of the scheme it is desirable that at each time step only a linear equation must be solved. This linear equation should be posed on the old domain. Our scheme will provide exactly this. As a by-product of our stability result in Theorem 2.2 we find a short-time existence result for (2.1)-(2.3) in Corollary 2.3. Such a result (in different function spaces) was proved earlier by Radkevich in [8]. Our approach is more elementary in the sense that it involves less functional-analytic machinery.