Asymptotic-numerical study of supersensitivity for generalized Burgers equations

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.