Summary
In many science and engineering problems, one observes smooth behaviour on macroscopic space and time scales. However, sometimes only a microscopic evolution law is known. In such cases, one can approximate the macroscopic time evolution by performing appropriately initialized simulations of the available microscopic model in small portions of the space-time domain. This coarse-grained time-stepper can be used to perform time-stepper based numerical bifurcation analysis. We discuss our recent results concerning the accuracy of the proposed methods.
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References
A. Bensoussan, J.L. Lions, and G. Papanicolaou. Asymptotic Analysis of Periodic Structures, volume 5 of Studies in Mathematics and its Applications. North-Holland, Amsterdam, 1978.
C.W. Gear and I.G. Kevrekidis. Constraint-defined manifolds: a legacy code approach to low-dimensional computation. Technical Report physics/0312094, arXiv e-Print archive, 2003.
I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg, and C. Theodoropoulos. Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis. Communications in Mathematical Sciences, 1(4):715–762, 2003.
K. Lust, D. Roose, A. Spence, and A. Champneys. An adaptive Newton-Picard algorithm with subspace iteration for computing periodic solutions. SIAM Journal on Scientific Computing, 19(4):1188–1209, 1998.
Y.H. Qian and S.A. Orszag. Scalings in diffusion-driven reaction A + B → C: Numerical simulations by Lattice BGK Models. Journal of Statistical Physics, 81(1/2):237–253, 1995.
G. Samaey, I.G. Kevrekidis, and D. Roose. Patch dynamics with buffers for homogenization problems. Technical Report physics/0412005, arXiv e-Print archive, 2004. Submitted to Journal of Computational Physics.
G. Samaey, D. Roose, and I.G. Kevrekidis. The gap-tooth scheme for homogenization problems. SIAM Multiscale Modeling and Simulation, 2004. In press.
C. Theodoropoulos, Y.H. Qian, and I.G. Kevrekidis. “Coarse” stability and bifurcation analysis using time-steppers: a reaction-diffusion example. Proceedings of the National Academy of Sciences, 97(18):9840–9843, 2000.
P. Van Leemput and K. Lust. Numerical bifurcation analysis of lattice Boltzmann models: a reaction-diffusion example. In M. Bubak, G.D. van Albada, P.M.A. Sloot, and J. Dongarra, editors, Computational Science — ICCS 2004, volume 3039 of LNCS, pages 572–579. Springer-Verlag, 2004.
P. Van Leemput, K. Lust, and I.G. Kevrekidis. Coarse-grained numerical bifurcation analysis of lattice Boltzmann models. Technical Report TW 410, Dept. of Computer Science, K.U.Leuven, 2004. Submitted to Physica D.
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Van Leemput, P., Samaey, G., Lust, K., Roose, D., Kevrekidis, I. (2006). Coarse-Grained Simulation and Bifurcation Analysis Using Microscopic Time-Steppers. In: Di Bucchianico, A., Mattheij, R., Peletier, M. (eds) Progress in Industrial Mathematics at ECMI 2004. Mathematics in Industry, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28073-1_96
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DOI: https://doi.org/10.1007/3-540-28073-1_96
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28072-9
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