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01.09.2017

Backward difference formulae for Kuramoto–Sivashinsky type equations

verfasst von: Georgios Akrivis, Yiorgos-Sokratis Smyrlis

Erschienen in: Calcolo | Ausgabe 3/2017

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Abstract

We analyze the discretization of the periodic initial value problem for Kuramoto–Sivashinsky type equations with Burgers nonlinearity by implicit–explicit backward difference formula (BDF) methods, establish stability and derive optimal order error estimates. We also study discretization in space by spectral methods.

Vorheriger Artikel Numerical validation for systems of absolute value equations

Nächster Artikel A new class of nonmonotone adaptive trust-region methods for nonlinear equations with box constraints

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Note that, if s is a nonnegative integer, then $||\, \cdot \, ||_{H^s}$ is equivalent to the norm defined by

$$\begin{aligned} ||u||_s = \left( \sum _{j=0}^s\int _0^L |u^{(j)}(x)|^2\,dx\right) ^{1/2}. \end{aligned}$$

Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, vol. 55 (1964)

Akrivis, G.: Stability of implicit–explicit backward difference formulas for nonlinear parabolic equations. SIAM J. Numer. Anal. 53, 464–484 (2015)MathSciNetCrossRefMATH

Akrivis, G., Crouzeix, M., Makridakis, Ch.: Implicit–explicit multistep methods for quasilinear parabolic equations. Numer. Math. 82, 521–541 (1999)MathSciNetCrossRefMATH

Akrivis, G., Kalogirou, A., Papageorgiou, D.T., Smyrlis, Y.-S.: Linearly implicit schemes for multi-dimensional Kuramoto–Sivashinsky type equations arising in falling film flows. IMA J. Numer. Anal. 36, 317–336 (2016)MathSciNetCrossRefMATH

Akrivis, G., Katsoprinakis, E.: Backward difference formulae: new multipliers and stability properties for parabolic equations. Math. Comp. 85, 2195–2216 (2016)MathSciNetCrossRefMATH

Akrivis, G., Lubich, C.: Fully implicit, linearly implicit and implicit–explicit backward difference formulae for quasi-linear parabolic equations. Numer. Math. 131, 713–735 (2015)MathSciNetCrossRefMATH

Akrivis, G., Papageorgiou, D.T., Smyrlis, Y.-S.: Linearly implicit methods for a semilinear parabolic system arising in two-phase flows. IMA J. Numer. Anal. 31, 299–321 (2011)MathSciNetCrossRefMATH

Akrivis, G., Papageorgiou, D.T., Smyrlis, Y.-S.: Computational study of the dispersively modified Kuramoto–Sivashinsky equation. SIAM J. Sci. Comp. 34, A792–A813 (2012)MathSciNetCrossRefMATH

Akrivis, G., Smyrlis, Y.-S.: Implicit–explicit BDF spectral methods for the Kuramoto–Sivashinsky equation. Appl. Numer. Math. 51, 151–169 (2004)MathSciNetCrossRefMATH

10.

Akrivis, G., Smyrlis, Y.-S.: Linearly implicit schemes for a class of dispersive–dissipative systems. Calcolo 48, 145–172 (2011)MathSciNetCrossRefMATH

11.

Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods in fluid dynamics. Springer series in computational physics. Springer, New York (1988)CrossRefMATH

12.

Constantin, P., Foias, C., Nicolaenko, B., Temam, R.: Integral manifolds and inertial manifolds for dissipative partial differential equations. Appl. Math. Sci., vol. 70, Springer, New York (1988)

13.

Crouzeix, M.: Une méthode multipas implicite–explicite pour l’approximation des équations d’évolution paraboliques. Numer. Math. 35, 257–276 (1980)MathSciNetCrossRefMATH

14.

Frankel, M., Roytburd, V.: Dissipative dynamics for a class of nonlinear pseudo-differential equations. J. Evol. Equ. 8, 491–512 (2008)MathSciNetCrossRefMATH

15.

Goodman, J.: Stability of the Kuramoto–Sivashinsky and related systems. Comm. Pure Appl. Math. 47, 293–306 (1994)MathSciNetCrossRefMATH

16.

Hairer, E., Wanner, G.: Solving ordinary differential equations II: Stiff and differential–algebraic problems. Springer series in computational mathematics, vol. 14, 2nd rev edn. Springer, Heidelberg (1991)

17.

Ioakim, X., Smyrlis, Y.-S.: Analyticity for a class of non-linear evolutionary pseudo-differential equations. Eur. J. Appl. Math. 25, 783–793 (2014)MathSciNetCrossRefMATH

18.

Kas-Danouche, S., Papageorgiou, D.T., Siegel, M.: A mathematical model for core-annular flows with surfactants. Divulg. Mat. 12, 117–138 (2004)MathSciNetMATH

19.

Papageorgiou, D.T., Maldarelli, C., Rumschitzki, D.S.: Nonlinear interfacial stability of core-annular film flow. Phys. Fluids A 2(3), 340–352 (1990)MathSciNetCrossRefMATH

20.

Kawahara, T.: Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation. Phys. Rev. Lett. 51, 381–382 (1983)CrossRef

21.

Kuramoto, Y.: Diffusion-induced chaos in reaction systems. Suppl. Prog. Theor. Phys. 64, 346–367 (1978)CrossRef

22.

Lubich, C., Mansour, D., Venkataraman, C.: Backward difference time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. 33, 1365–1385 (2013)MathSciNetCrossRefMATH

23.

Nevanlinna, O., Odeh, F.: Multiplier techniques for linear multistep methods. Numer. Funct. Anal. Optim. 3, 377–423 (1981)MathSciNetCrossRefMATH

24.

Sivashinsky, G.I.: On flame propagation under conditions of stoichiometry. SIAM J. Appl. Math. 39, 67–82 (1980)MathSciNetCrossRefMATH

25.

Stanislavova, M., Stefanov, A.: Asymptotic estimates and stability analysis of Kuramoto–Sivashinsky type models. J. Evol. Equ. 11, 605–635 (2011)MathSciNetCrossRefMATH

26.

Tadmor, E.: The well-posedness of the Kuramoto–Sivashinsky equation. SIAM J. Math. Anal. 17, 884–893 (1986)MathSciNetCrossRefMATH

Titel: Backward difference formulae for Kuramoto–Sivashinsky type equations
verfasst von: Georgios Akrivis
Yiorgos-Sokratis Smyrlis
Publikationsdatum: 01.09.2017
Verlag: Springer Milan
Erschienen in: Calcolo / Ausgabe 3/2017
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-016-0205-0

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