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This paper examines a linear extrapolation time-discretization of a $2D$ phase relaxation model with temperature dependent convection and reaction. The model consists of a diffusion-advection PDE for temperature and an ODE with double obstacle $\pm 1$ for phase variable. Under a stability constraint, this scheme is shown to converge with optimal orders $\mathcal{O}\left(\tau {\left|\log \tau \right|}^{1/2}\right)$ for temperature and enthalpy, and $\mathcal{O}\left({\tau }^{1/2}{\left|\log \tau \right|}^{1/2}\right)$ for heat flux as time-step $\tau \downarrow 0$.