Dahlquist's barriers and much beyond

On the occasion of 60 years to a seminal paper by the Swedish mathematician Germund Dahlquist, this review paper starts by discussing the celebrated Dahlquist's barriers, which are theoretical limitations on the accuracy and stability properties of a broad and important class of time-stepping methods, i.e., Linear Multistep (LMS) methods, for the solution of initial value problems. Perhaps the two most dramatic Dahlquist barriers are the one that precludes an explicit LMS method from being unconditionally stable, and the one that precludes a high-order LMS method from being unconditionally stable. We discuss Dahlquist's barriers and also later barriers proved by other authors. We then discuss some time-stepping methods which seemingly break at least one of Dahlquist's barriers. Of course, the explanation of this “paradox” is that these are not LMS methods, so the barriers do not necessarily apply to them. We relate to two types of barrier breakers: those that break stability barriers (explicit unconditionally-stable methods), and those that break order barriers (high-order explicit or unconditionally-stable methods). We also review some current advanced time-integration techniques which significantly deviate from the LMS format.