document: Three Runge–Kutta IMEX schemes were tested by Ullrich and Jablonowski [23] for the HEVI solution of the equations governing atmospheric motion. They tested the ARS(2,3,2) scheme of Ascher et al. [1] and also suggested the less computationally expensive but nearly as accurate Strang carryover scheme. This involves Strang splitting but the first implicit stage is cleverly re-used from the final implicit stage of the previous time-step and so there is only one implicit solution per time-step. Another novel approach taken by Ullrich and Jablonowski [23] is to use a Rosenbrock solution in order to treat all of the vertical terms implicitly rather than just the terms involved in wave propagation. A Rosenbrock solution is one iteration of a Newton solver. This circumvents the time-step restriction associated with vertical advection at the cost of slowing the vertical advection.
extractive_keyphrases: atmospheric motion hevi solution newton solver rosenbrock solution runge–kutta imex schemes slowing the vertical advection strang carryover scheme strang splitting vertical advection wave propagation
abstractive_keyphrases: ars(2,3,2) scheme one iteration of a newton solver time-step restriction