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| Main Authors: | , |
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| Format: | Preprint |
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1999
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| Online Access: | https://arxiv.org/abs/math/9908051 |
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| _version_ | 1866912655900409856 |
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| author | Garbey, Marc Kaper, Hans G. |
| author_facet | Garbey, Marc Kaper, Hans G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_9908051 |
| institution | arXiv |
| publishDate | 1999 |
| record_format | arxiv |
| spellingShingle | Asymptotic-numerical study of supersensitivity for generalized Burgers equations Garbey, Marc Kaper, Hans G. Numerical Analysis Dynamical Systems 35B25, 35B30 (Primary) 35Q53, 65M55 (Secondary) 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. |
| title | Asymptotic-numerical study of supersensitivity for generalized Burgers equations |
| topic | Numerical Analysis Dynamical Systems 35B25, 35B30 (Primary) 35Q53, 65M55 (Secondary) |
| url | https://arxiv.org/abs/math/9908051 |