This paper is devoted to the numerical computation of a steady-state thermo-fluid modeling related to the Friction Stir Welding Process in a two-dimensional cylindrical geometry. It analyzes the efficiency of the implementation on parallel architectures of two finite-difference schemes on a structured grid. The first one applies the Newton-Raphson method to compute a numerical solution of this non-linear elliptic type equation, and uses an iterative sparse solver. The second one is based on a time-marching approach converging to the steady state solution thanks to a time-explicit computation. Their respective performance is presented and discussed. Some numerical simulation results are presented to validate the proposed approach.