Numerical solutions to a BBM-Burgers model with a nonlocal viscous term

Serge Dumont (1, 2) , Imen Manoubi (3)

1 UNIMES - Université de Nîmes
2 IMAG - Institut Montpelliérain Alexander Grothendieck
3 Département de Mathématiques [Monastir]

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.

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Serge Dumont, Imen Manoubi. Numerical solutions to a BBM-Burgers model with a nonlocal viscous term. Numerical Methods for Partial Differential Equations, 2018, 34 (6), pp.2279-2300. ⟨10.1002/num.22291⟩. ⟨hal-01818121⟩