dorsal/arxiv
View SchemaThe gap-tooth scheme for homogenization problems
| Authors | Giovanni Samaey, Dirk Roose, Ioannis G. Kevrekidis |
|---|---|
| Categories | |
| ArXiv ID | physics/0312004 |
| URL | https://arxiv.org/abs/physics/0312004 |
Abstract
An important class of problems exhibits smooth behaviour in space and time on a macroscopic scale, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, an ``equation-free framework'' has been proposed, of which the gap-tooth scheme is an essential component. The gap-tooth scheme is designed to approximate a time-stepper for an unavailable macroscopic equation in a macroscopic domain; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes, which cover only a fraction of the domain. We analyze the convergence of this scheme for a parabolic homogenization problem with non-linear reaction. In this case, the microscopic model is a partial differential equation with rapidly oscillating coefficients, while the unknown macroscopic model is approximated by the homogenized equation. We show that our method approximates a finite difference scheme of arbitrary (even) order for the homogenized equation when we appropriately constrain the microscopic problem in the boxes. We illustrate this theoretical result with numerical tests on several model problems. We also demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffer regions around the computational boxes.
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"date_created": "2026-03-02T18:00:47.022000Z",
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"abstract": "An important class of problems exhibits smooth behaviour in space and time on\na macroscopic scale, while only a microscopic evolution law is known. For such\ntime-dependent multi-scale problems, an ``equation-free framework\u0027\u0027 has been\nproposed, of which the gap-tooth scheme is an essential component. The\ngap-tooth scheme is designed to approximate a time-stepper for an unavailable\nmacroscopic equation in a macroscopic domain; it uses appropriately initialized\nsimulations of the available microscopic model in a number of small boxes,\nwhich cover only a fraction of the domain. We analyze the convergence of this\nscheme for a parabolic homogenization problem with non-linear reaction. In this\ncase, the microscopic model is a partial differential equation with rapidly\noscillating coefficients, while the unknown macroscopic model is approximated\nby the homogenized equation. We show that our method approximates a finite\ndifference scheme of arbitrary (even) order for the homogenized equation when\nwe appropriately constrain the microscopic problem in the boxes. We illustrate\nthis theoretical result with numerical tests on several model problems. We also\ndemonstrate that it is possible to obtain a convergent scheme without\nconstraining the microscopic code, by introducing buffer regions around the\ncomputational boxes.",
"arxiv_id": "physics/0312004",
"authors": [
"Giovanni Samaey",
"Dirk Roose",
"Ioannis G. Kevrekidis"
],
"categories": [
"physics.comp-ph"
],
"title": "The gap-tooth scheme for homogenization problems",
"url": "https://arxiv.org/abs/physics/0312004"
},
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