, Volume 48, Issue 2, pp 145–172

Linearly implicit schemes for a class of dispersive–dissipative systems


DOI: 10.1007/s10092-010-0033-6

Cite this article as:
Akrivis, G. & Smyrlis, YS. Calcolo (2011) 48: 145. doi:10.1007/s10092-010-0033-6


We consider initial value problems for semilinear parabolic equations, which possess a dispersive term, nonlocal in general. This dispersive term is not necessarily dominated by the dissipative term. In our numerical schemes, the time discretization is done by linearly implicit schemes. More specifically, we discretize the initial value problem by the implicit–explicit Euler scheme and by the two-step implicit–explicit BDF scheme. In this work, we extend the results in Akrivis et al. (Math. Comput. 67:457–477, 1998; Numer. Math. 82:521–541, 1999), where the dispersive term (if present) was dominated by the dissipative one and was integrated explicitly. We also derive optimal order error estimates. We provide various physically relevant applications of dispersive–dissipative equations and systems fitting in our abstract framework.


Dispersive–dissipative systems Semilinear parabolic equations Kuramoto–Sivashinsky equation Linearly implicit schemes Backward differentiation formulas (BDF) methods Error estimates 

Mathematics Subject Classification (2000)

65M60 65M12 65L06 

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Computer Science DepartmentUniversity of IoanninaIoanninaGreece
  2. 2.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus

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