Advantage of multi steps ODE methods over single step methods

Question

I wanna know what's the advantage of multi step ODE methods such as Adams-Bashforth over ordinary single step methods such Runge–Kutta, accuracy/time wise.

Answer 1

The main advantage of explicit multistep methods compared to explicit Runge-Kutta lies in the fact that adding additional steps/function evaluations results (for reasonable multi-step) always in increase of the order of accuracy.

The classic examples thereof include the (explicit) Adams-Bashforth family, for which the the order of consistency $p$ is equal to the number of steps $k$ and the (explicit) Nyström methods with $p=k, k > 2$. [1]

This is in stark contrast to explicit Runge-Kutta methods, for which the well-known Butcher-Barrier [2], [3] kicks in at order 5, which states for orders $p \geq 5$ one needs at least $s \geq p + 1$ stages. This is innocent looking, but until now, there is no 7'th order accurate method with 8 stages known - the best known method uses 9 stages. This seems to become more severe for higher orders, as the best currently 8'th order explicit RK method requires 11 stages [4].

Regarding implicit methods, the Butcher Barrier does not exist - in fact, there exist implicit methods with order $2s$ for all $s$ [5], [6], [1]. Implicit multistep methods are much less attractive given that they are, in contrast to many implicit Runge-Kutta methods, not A-stable for orders above 2 (Second Dahlquist barrier [7]) and the order of consistency scales for most methods (e.g. Adams-Moulton, Backwards-Differentiation-Formulas(BDF)) only with $k+1$ instead of $2k$ as for some RK methods [1]. These facts make them much less attractive compared to e.g. Gauss-Legendre methods.

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It should be mentioned that there are of course many more considerations that one could make, e.g. the domains of absolute stability of explicit multistep and Runge-Kutta methods, storage efficiency, ease of varying timesteps, internal stability properties, combination of explicit and implicit methods, and so on.

But all of this would go far beyond a MSE post. If you want to dive deep, consider the books by Hairer & Wanner [1], [9], the book by Butcher [4] or the one by Iserles [10].