Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions

• Autores: Acosta, G.; Bonder, J.F.; Groisman, P.; Rossi, J.D.
• Año de publicación: 2002
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: Math. Model. Numer. Anal. 2002;36(1):55-68
• Materia: Asymptotic behavior; Blow-up; Non-linear boundary conditions; Parabolic equations; Semi-discretization in space
• Institución: Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales

We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω × (0, T); fully coupled by the boundary conditions ∂u/∂η = up11vp12, ∂v/∂η = up21vp22 on ∂Ω × (0, T), where Ω is a bounded smooth domain in ℝd. 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.
Fil:Acosta, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fil:Groisman, P. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.