In this paper, we numerically investigate the BBM-Burgers equation with a nonlocal viscous term $$u_t+u_x+\beta u_{xxx}+\sqrt{\nu}D^{1/2}u+\gamma u u_x=\alpha u_{xx}$$ where $$ D^{1/2}u(t)=\frac{1}{\sqrt{\pi}}\frac{\partial}{\partial t}\int_0^t \frac{u(s)}{\sqrt{t-s}}\mathrm{d}s $$ is the Riemann-Liouville half-order derivative in time. In particular, we implement different numerical schemes to approximate the solution and its asymptotical behavior. Also, we compare our numerical results with those given in 2013, 2014 for similar models.