广义Burgers方程超灵敏性的渐近-数值研究
Asymptotic-numerical study of supersensitivity for generalized Burgers equations
AI总结：
本文针对广义Burgers方程，研究边界扰动对内部过渡层位置的超灵敏影响，提出计算稳态过渡层位置的算法，该算法可推广至凸非线性粘性守恒律并支持并行计算。
摘要：
本文研究了Burgers方程$-εu_{xx} + u_t + u u_x = 0$在$(-1,1)$上，边界条件$u(-1) = 1 + δ$，$u(1) = -1$，以及其在二维上的推广$-εΔu + u_t + u u_x + u u_y = 0$在$(-1,1) \times (-π, π)$上，边界条件$u|_{x=1} = 1 + δ$，$u|_{x=-1} = -1$，且在$y$方向具有$2π$周期性的解的一些渐近和数值问题。扰动参数$δ$和$ε$是任意小的正数且相互独立；当它们趋近于0时，满足渐近阶关系$δ= O_s ({\rm e}^{-a/ε})$，其中$a \in (0,1)$为常数。这些对流主导的粘性守恒律的解在区域内部表现出一个过渡层，其位置在$t\to∞$时对边界扰动具有超灵敏性。本文提出了计算稳态时过渡层位置的算法。这些算法推广到具有凸非线性的粘性守恒律，并在并行计算环境中具有可扩展性。
英文摘要：
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.
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