Linearly implicit schemes for a class of dispersive-dissipative systems

Abstract

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. © 2010 Springer-Verlag.