Summary
An important class of problems exhibits macroscopically smooth behaviour in space and time, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, the gap-tooth scheme has recently been proposed. The scheme approximates the evolution of an unavailable (in closed form) macroscopic equation in a macroscopic domain; it only uses appropriately initialized simulations of the available microscopic model in a number of small boxes. For some model problems, including numerical homogenization, the scheme is essentially equivalent to a finite difference scheme, provided we define appropriate algebraic constraints in each time-step to impose near the boundary of each box. Here, we demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffers that “shield” over relatively short time intervals the dynamics inside each box from boundary effects. We explore and quantify the behavior of these schemes systematically through the numerical computation of damping factors of the corresponding coarse time-stepper, for which no closed formula is available.
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References
W. E and B. Engquist. The heterogeneous multi-scale methods. Comm. Math. Sci.1(1):87–1322003.
C.W. Gear and I.G. Kevrekidis. Boundary processing for Monte Carlo simulations in the gap-tooth scheme. physics/0211043 at arXiv.org, 2002.
C.W. Gear, I.G. Kevrekidis, and C. Theodoropoulos. “Coarse” integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods. Comp. Chem. Eng.26(7-8):941–9632002.
C.W. Gear, J. Li, and I.G. Kevrekidis. The gap-tooth method in particle simulations. Physics Letters A316:190–1952003.
N. G. Hadjiconstantinou. Hybrid atomistic-continuum formulations and the moving contact-line problem. J. Comp. Phys.154:245–2651999.
T.Y. Hou and X.H. Wu. A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comp. Phys.134:169–1891997.
I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg, and C. Theodoropoulos. Equation-free multiscale computation: enabling microscopic simulators to perform system-level tasks. Comm. Math. Sci. Submitted, physics/0209043 at arxiv.org, 2002.
J. Li, D. Liao, S. Yip. Imposing field boundary conditions in MD simulation of fluids: optimal particle controller and buffer zone feedbackMat. Res. Soc. Symp. Proc538(473-478)1998.
G. Samaey, I.G. Kevrekidis, and D. Roose. The gap-tooth scheme for homogenization problems. SIAM MMS2003. Submitted.
S. Setayeshar, C.W. Gear, H.G. Othmer, and I.G. Kevrekidis. Application of coarse integration to bacterial chemotaxis. SIAM MMS. Submitted, physics/0308040 at arxiv.org, 2003.
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© 2004 Springer-Verlag Berlin Heidelberg
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Samaey, G., Kevrekidis, I.G., Roose, D. (2004). Damping factors for the gap-tooth scheme. In: Attinger, S., Koumoutsakos, P. (eds) Multiscale Modelling and Simulation. Lecture Notes in Computational Science and Engineering, vol 39. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18756-8_6
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DOI: https://doi.org/10.1007/978-3-642-18756-8_6
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