Domain decomposition method to realize an implicit difference scheme for the one-phase Stefan problem

A new algorithm is proposed for the approximate realization of an implicit difference scheme for the one-phase Stefan problem formulated in the form of a variational inequality. This algorithm is theoretically based on the estimates of movement rate for the phase transition mesh boundary and on the estimates of the boundary perturbation impact on the solution of a difference scheme. The approximate realization of a difference scheme consists in its replacement by two problems in overlapping mesh subdomains, one of them being a linear mesh problem and, the other, a nonlinear problem. To solve each of these problems, a method is then proposed to decompose the domain into small subdomains. The Stefan problem and numerical methods of its solution are investigated in a large number of publications (see [5-7, 11-13] and the references therein). Many authors investigated, among other things, the convergence and accuracy of difference schemes. This paper treats a very simple implicit difference scheme on a uniform mesh for the one-phase Stefan problem. To find its solution at each time step, it is necessary to solve a nonlinear problem, namely, a mesh obstacle problem. The paper suggests a new method of the approximate solution of this problem, which is essentially as follows. The original mesh domain is decomposed into a set of overlapping subdomains of small dimensions. In the subdomains, mesh problems are solved independently. The linear difference schemes constitute the larger part of these problems, while mesh obstacle problems, the smaller part. The linear problems can be solved by efficient direct or iterative methods. To solve nonlinear problems, one can use different iterative methods [7], the splitting method [9,10] in particular. In Section 1, we formulate the one-phase Stefan problem in the form of a variational inequality and discuss the known properties of its solution. In Section 2, we construct a difference scheme and investigate the movement rate for the phase transition mesh boundary. We establish here the estimate of the maximal distance at which this mesh boundary can be moved in one time step. In Section 3, we carry out an approximate splitting of the nonlinear mesh problem for a fixed time step into a linear and a nonlinear subproblems in two overlapping mesh subdomains. In this case, the mesh subdomain for the nonlinear subproblem is located in the neighbourhood of the mesh boundary of the phase transition from the preceding time step. In Section 4, we consider the nonlinear mesh subproblem. To solve it, following [8] we use an iteration-free method of decomposing a domain into small subdomains and give its justification. In Section 5, we outline the method and consider certain implementation aspects. 488 Yu. A. Kuznetsov and A. V. Lapin 1. A VARIATIONAL FORMULATION OF THE ONE-PHASE STEFAN PROBLEM Let Ω = (0,1) χ (0,1) be the unit square with the boundary Γ = (Jf=1Fi, where Γ1 = [xi = 0}, F2 = (x2 = 0} u (x2 = 1}, F3 = {xl = 1}. We seek functions 0(x, t) and /(x) satisfying the following relations (see [5,13] and others): -—Δθ = 0, χεΩ, t>l(x) dt 0 = 0, χθΩ, t^l(x) V0.V/=-/c, Χ6Ω, ί = /(χ) 0 = 0! (χ2), xeF1} ί > 0 (1.1) 0 = 0, xeF3, ί > 0 de Λ ^ — = 0, xeF2, ί > 0 0(x,0) = 0, Χ6Ω where η is the vector of the normal to Ω, θt is a prescribed positive function, and k = const > 0. The problem formulated is a mathematical model of the process of melting of ice with zero temperature, which occupies the domain Ω at the time moment t = 0. Here, θ — 0(x, t) is the temperature, t = /(x) is the equation for the boundary of the water-ice phase transition, k is the specific melting heat. At every time moment 0 < ί < T(for T > 0) the domain Ω consists of two subdomains: Ω+(ί) = {χεΩ| θ(χ, ί) > 0} occupied by water Ω0(ί) = {χθΩ| 0(x, t) = 0} occupied by ice. The boundary 3Ω+(ί)ηΩ separating these subdomains is the phase transition boundary. Following [4], define the function Ρ 0(*, Jl(x) "(*' = > ο, ,</(*). τ) dr, t > /(x) (1.2) Assume that 9leW\l(r]}r^C(Yl\ and define V = {veH(Si)\ v(x) = 0 on F3} K(t) = {veV\v(x)^QmQ,, v(x) = 61(x)t on FJ. (1.3) The function Μ is a solution to the following variational inequality: for almost all ie[0, T]: u(t)eK(t) and