Asymptotic-Numerical Study of Supersensitivity for Generalized Burgers' Equations

This article addresses some asymptotic and numerical issues related to the solution of Burgers’ equation, −εuxx+ut+uux = 0 on (−1, 1), subject to the boundary conditions u(−1) = 1+δ, u(1) = −1, and its generalization to two dimensions, −ε∆u+ut+uux+uuy = 0 on (−1, 1)×(−π, π), subject to the boundary conditions u|x=1 = 1 + δ, u|x=−1 = −1, with 2π periodicity in y. The perturbation parameters δ and ε are arbitrarily small positive and independent; when they approach 0, they satisfy the asymptotic order relation δ = Os(e) for some constant a ∈ (0, 1). The solutions of these convection-dominated viscous conservation laws exhibit a transition layer in the interior of the domain, whose position as t → ∞ is supersensitive to the boundary perturbation. Algorithms are presented for the computation of the position of the transition layer at steady state. The algorithms generalize to viscous conservation laws with a convex nonlinearity and are scalable in a parallel computing environment. AMS subject classifications. Primary 35B25, 35B30; Secondary 35Q53, 65M55