Robustness of implicit Runge-Kutta schemes with respect to errors in the specification of the system's complex frequencies

The note uses the idealized single-mode complex-frequency response analysis to investigate (subject to the limitations inherent in this idealization) the stability and robustness of some time integration schemes that might be considered candidates for incorporation within a model where both long time steps and a high order of formal accuracy are desired. Generally, we would expect that such models would be formulated in the semi-Lagrangian style, although this is not strictly necessary. The family of schemes we consider can broadly be categorized as 'implicit Runge- Kutta' (of which the trapezoidal, and decentered generalizations are relatively trivial examples). The numerical robustness of these implicit schemes under simplified conditions depends upon the number of applications of the imperfect corrector iterations, and the amounts of decentering at each stage. The idealized experiments we describe shed light on this dependency and narrow down the choices we might wish to consider in formulating a formally accurate numerical model with a fuller degree of implicitness than is customarily adopted within the semi-Lagrangian paradigm. [doi:10.7289/V59Z92XW (http://dx.doi.org/10.7289/V59Z92XW)] R. James Purser. "May 19, 2016." "This is an unreviewed manuscript, primarily intended for informal exchange of information among the NCEP staff members." System requirements: Adobe Acrobat Reader. Includes bibliographical references (pages 23-24). 2016 NWS (National Weather Service) NCEP (National Centers for Environmental Prediction) Library http://doi.org/10.7289/V59Z92XW Public Domain 2285