  This article addresses some asymptotic and numerical issues related to the
solution of Burgers' equation, $-\epsilon u_{xx} + u_t + u u_x = 0$ on
$(-1,1)$, subject to the boundary conditions $u(-1) = 1 + \delta$, $u(1) = -1$,
and its generalization to two dimensions, $-\epsilon \Delta u + u_t + u u_x + u
u_y = 0$ on $(-1,1) \times (-\pi, \pi)$, subject to the boundary conditions
$u|_{x=1} = 1 + \delta$, $u|_{x=-1} = -1$, with $2\pi$ periodicity in $y$. The
perturbation parameters $\delta$ and $\epsilon$ are arbitrarily small positive
and independent; when they approach 0, they satisfy the asymptotic order
relation $\delta = O_s ({\rm e}^{-a/\epsilon})$ for some constant $a \in
(0,1)$.
  The solutions of these convection-dominated viscous conservation laws exhibit
a transition layer in the interior of the domain, whose position as
$t\to\infty$ 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.
