Efficient iterative linearised solvers for numerical approximations of stochastic Stefan problems

TL;DR

This paper introduces efficient iterative linearised solvers for numerical schemes solving stochastic Stefan problems, emphasizing computational advantages and convergence considerations in stochastic and deterministic contexts.

Contribution

It proposes linearised iterative solvers that reuse the same coefficient matrix across time steps and realizations, improving efficiency in solving stochastic Stefan problems.

Findings

Linearised solvers reduce computational cost by reusing matrix factorizations.

Sensitivity analysis guides optimal solver choice in different scenarios.

Adaptive tolerance improves solver performance in stochastic and deterministic cases.

Abstract

We present iterative solvers to approximate the solution of numerical schemes for stochastic Stefan problems. After briefly talking about the convergence results, we tackle the question of efficient strategies for solving the nonlinear equation associated with this scheme. We explore several approaches, from a standard Newton technique to linearised solvers. The latter offer the advantage of using the same coefficient matrix of the linearised system in each nonlinear iteration, for all time steps, and across all realisations of the Brownian motions. As a consequence, the system can be factorised once and for all. Although the linearised approach has a slower convergence rate, our sensitivity analysis and the use of adaptive tolerance in both deterministic and stochastic cases provide valuable insights for choosing the most effective solver across various scenarii.