This paper presents the behavior of general parallel synchronous and asynchronous multisplitting and two-stage methods for the numerical simulation of steel solidification in continuous casting. Thanks to the mathematical analysis and the implementation of these methods one can show the results of parallel experiments for the target application. The mathematical model is constituted by coupled nonlinear boundary value problems, namely the heat equation taking into account, on part of the boundary, a radiation phenomenon described by the Stefan law. For the numerical solution of such partial differential equations we consider, depending on whether the coefficient of thermal conductivity is constant or temperature-dependent, both an implicit or a semi-implicit discretization with respect to the time of the studied evolution problem, while the spatial discretization is carried out by adapted finite difference schemes. Then large scale discretized algebraic systems are solved by sequential and synchronous or asynchronous iterative algorithms; comparison of these various previous methods implemented on clusters and grid are achieved in both cases when the thermal conductivity is constant and more generally dependent of the temperature.