Incompressible, convective-dominant, thermally coupled, non-Newtonian flows with phase change are critical in industrial processes and natural phenomena. The mathematical modeling of these problems involves nonlinearities such as convective acceleration, temperature-dependent thermodynamic properties, energy balance at the liquid-solid moving boundary, and thermal coupling. The most common alternative to linearize these problems and reduce the computational cost of simulations is an iterative Picard fixed-point strategy. However, these techniques often suffer from slow convergence, increasing computation times. This work proposes a Picard-Anderson accelerated scheme to improve the slow convergence ratio of a classical Picard strategy. Previous studies on acceleration methods in fluid and heat transfer coupled problems have focused on Newtonian steady state cases [1]. In this presentation, the Anderson Acceleration method, combined with the Finite Volume Method (FVM), is applied to solve unsteady thermally coupled phase change problems efficiently. Including novel numerical techniques will reduce the computational time of simulations by accelerating the convergence rate of iterative cycles. The Anderson Acceleration method in the FVM is implemented on collocated meshes in two dimensions using an in-house developed code [2]. The accelerated FVM is evaluated in forced convection, natural convection and water freezing problems. The main results indicate that the Anderson Acceleration method reduces the number of iterations of nonlinearity cycles in thermally coupled problems, reinforcing Picard's capability to converge to solutions in highly challenging non-Newtonian flows.