Numerical difference solution of moving boundary random Stefan problems

Abstract

[EN] This paper deals with the construction of numerical solutions of moving boundary random problems where the uncertainty is limited to a finite degree of randomness in the mean square framework. Using a front fixing approach the problem is firstly transformed into a fixed boundary one. Then a random finite difference scheme for both the partial differential equation and the Stefan condition, allows the discretization. Since statistical moments of the approximate stochastic process solution are required, we combine the sample approach of the difference schemes together with Monte Carlo technique to perform manageable approximations of the expectation and variance of both the approximating stochastic process solution and the stochastic moving boundary solution. Qualitative and reliability properties such as positivity, monotonicity and stability in the mean square sense are treated. Feasibility of the proposed method is checked with illustrative examples of a melting problem and a binary metallic alloys problems.