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Galerkin modelling of the Burgers equation using harmonic wavelets. (English) Zbl 1044.65511

Summary: In this paper we present a wavelet Galerkin approximation of the one-dimensional Burgers equation using complex harmonic wavelets. We derive exact expressions for the connection coefficients for the linear dissipation and nonlinear advection terms. A large reduction in the number of space-scale nonlinear modes is obtained by exploiting the band-limited spectra and localisation properties of harmonic wavelets. We discuss the advantages and disadvantages of implementing harmonic wavelets as opposed to the more standard choice of wavelets basis, highlighting the relevance to shell models of turbulence.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] (Benedetto, J. J.; Frasier, M. W., Wavelets: Mathematics and Applications (1994), CRC Press: CRC Press Boca Raton) · Zbl 0840.00013
[2] Newland, D. E., (Proc. R. Soc. Lond. A, 443 (1993)), 203 · Zbl 0793.42020
[3] Desnyanski, V. N.; Novikov, E. A., Prikl. Math. Mekh., 38, 507 (1974)
[4] Siggia, E., Phys. Rev. A, 15, 1730 (1977)
[5] Zimin, V. D., Izv. Akad. Nauk. SSSR Fiz. Atmos. Okeana, 17, 941 (1981)
[6] Mallat, S., Trans. Am. Math. Soc., 315, 69 (1989) · Zbl 0686.42018
[7] Daubechies, I., (Ten Lectures on Wavelets (1992), SIAM: SIAM Philadelphia) · Zbl 0776.42018
[8] Burgers, J. M., (Proc. R. Neth. Acad. Sci. Amsterdam, 17 (1940)), 2
[9] McComb, W. D., (The Physics of Fluid Turbulence (1990), Clarendon: Clarendon Oxford) · Zbl 0748.76005
[10] Meyer, Y., (Wavelets and Operators (1992), Cambridge: Cambridge Cambridge) · Zbl 0776.42019
[11] Davies, A. J., (The Finite Element Methods: A First Approach (1980), Clarendon: Clarendon Oxford) · Zbl 0433.65061
[12] Beylkin, G.; Keiser, J. M., J. Comp. Phys., 132, 233 (1997) · Zbl 0880.65076
[13] Bouchard, J. P.; Mezard, M.; Parisi, G., Phys. Rev. E, 52, 3656 (1995)
[14] Mouri, H.; Kubotani, H., Phys. Lett. A, 201, 53 (1995) · Zbl 1020.42500
[15] R. Benzi, L. Biferale, E. Trovatore, Preprint (1997).; R. Benzi, L. Biferale, E. Trovatore, Preprint (1997).
[16] Eggers, J.; Grossmann, S., Phys. Lett. A, 156, 444 (1991)
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