Abstract We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where separation scales in additive form, typically used implicit-explicit (IMEX) methods, is not possible. As shown Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta these techniques give a great flexibility, and allow, many cases, simple linearly impli...

We define a family of discrete Advection-reaction operators, denoted by Aaλ, which associate to a given scalar sequence s = {sn} the sequence given by Aaλ(s) ≡ {bn}, where bn = an−2sn−1 + λnsn for n = 1, 2, . . .. For Aaλ we explicitly find their iterates and study their convergence properties. Finally, we show the relationship between the family of discrete operators with the continuous one di...

In the late 1990s Allan Robinson developed a theory of NPZ interactions in a laminar upwelling flow field. His approach was to use the advection reaction (AR) equation in a Lagrangian coordinate system. Recently, his theory was extended to turbulent flow by applying a probability density function to the solution of the AR equation. A review of this work is presented as well as new work examinin...