Simultaneousvs.non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
Gabriel Acosta, Julián Fernández Bonder, Pablo Groisman, Julio Daniel Rossi · ESAIM Mathematical Modelling and Numerical Analysis · 2002
We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω x (0,T); fully coupled by the boundary conditions , on ∂Ω x (0,T), where Ω is a bounded smooth domain in . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation (U,V). We prove that if U blows up in finite time then V can fail to blow up if and only if p11 > 1 and p21 < 2(p11 - 1) , which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover, we find that if the continuous problem has non-simultaneous blow-up then the same is true for the discrete one. We also prove some results about the convergence of the scheme and the convergence of the blow-up times.