Asymptotic-numerical study of supersensitivity for generalized Burgers equations

TL;DR

Addresses the supersensitive dependence of the steady-state transition-layer position in convection-diffusion Burgers-type problems with vanishing viscosity; develops asymptotic insight and scalable numerical algorithms (domain-decomposition, adaptive Chebyshev-collocation, and profile-based long-time integration) to locate the layer with high accuracy; shows that in 1D the layer center $x^*_{\\infty}$ depends supersensitively on $\\delta$ with leading order $x^*_{\\infty} \\approx 1 - \\varepsilon \\ln(2/\\delta)$ when $\\delta = O_s(e^{-a/\\varepsilon})$; extends the analysis to 2D where the $y$-averaged layer position is supersensitive but the layer remains planar, and demonstrates scalable parallel algorithms that achieve accurate predictions for long-time simulations.

Abstract

This article addresses some asymptotic and numerical issues related to the solution of Burgers' equation, $-εu_{xx} + u_t + u u_x = 0$ on $(-1,1)$, subject to the boundary conditions $u(-1) = 1 + δ$, $u(1) = -1$, and its generalization to two dimensions, $-εΔu + u_t + u u_x + u u_y = 0$ on $(-1,1) \times (-π, π)$, 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 $δ= O_s ({\rm e}^{-a/ε})$ 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.

Figures (4)

• Figure 1: The profile function $u_0$ (dashed line) and the computed solution $U(\cdot\,,t)$ at some time $t$ (solid line); $\varepsilon=0.05$, $\delta = 1 \cdot 10^{-3}$.
• Figure 2: Contour lines of the solution $U$ at steady state.
• Figure 3: The solution $U$ at steady state.
• Figure 4: The difference $U - u_0$ at steady state.