Dahlquist's barriers
/ DAHL-kvist /
It would be lovely if you could just keep cranking up the order of a multistep method, or build one that is both high order AND rock-solid stable for stiff problems. Dahlquist proved you cannot — there are hard mathematical ceilings on what a linear multistep method can achieve. His two barriers are the fundamental no-go theorems that shape the whole design of multistep solvers.
The FIRST Dahlquist barrier limits accuracy-with-stability: a zero-stable linear k-step method can have order at most k + 1 if k is odd, and at most k + 2 if k is even (and a method reaching k + 2 has a stability boundary so tight it is useless in practice). In short, you cannot make a multistep method arbitrarily high order while keeping it convergent — stability fights accuracy, and order roughly tracks the number of steps. The SECOND Dahlquist barrier limits stiff capability: NO A-stable linear multistep method can have order greater than 2, and among second-order A-stable methods the trapezoidal rule has the smallest error constant. This is precisely why backward Euler (order 1) and BDF2 (order 2) are the only A-stable BDFs, and why high-order stiff solvers must settle for the weaker A(alpha)-stability.
These barriers matter because they explain WHY the landscape of methods looks the way it does. They are why Runge-Kutta methods (which the barriers do NOT constrain, because they are not linear multistep methods) are the route to high order with good stability — implicit Runge-Kutta methods can be A-stable at arbitrarily high order, sidestepping the second barrier entirely. The barriers are not engineering shortcomings to be fixed by a cleverer formula; they are proven impossibilities within the linear-multistep class, and recognizing them saves you from chasing a method that provably cannot exist.
Want a multistep method that is A-stable AND fourth order? The second barrier proves it cannot exist — so stiff solvers either accept order 2 (backward Euler, BDF2, trapezoidal) or drop to A(alpha)-stability (BDF3-6, which forfeit a wedge near the imaginary axis). To get A-stable high order you must leave the multistep class entirely and use an implicit Runge-Kutta method, which the barriers do not touch.
Hard ceilings: order vs zero-stability (first), and A-stability capped at order 2 (second).
The barriers apply ONLY to linear multistep methods. Runge-Kutta methods are exempt — implicit RK can be A-stable at any order — which is the deep reason both method families coexist rather than one obsoleting the other.