Asymptotic-numerical study of supersensitivity for generalized Burgers' equations

This article addresses some asymptotic and numerical issues related to the solution of Burgers' equation, -εu_xx+u_t+uu_x = 0 on (-1, 1), subject to the boundary conditions u(-1) = 1+δ, u(1) = -1, and its generalization to two dimensions, -εΔu+u_t+uu_x+uu_y = 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 = O_s(e^-a/ε) 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.