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The first Dahlquist barrier states that a zero-stable and linear q-step multistep method cannot attain an order of convergence greater than q + 1 if q is odd and greater than q + 2 if q is even. If the method is also explicit, then it cannot attain an order greater than q (Hairer, Nørsett & Wanner 1993, Thm III.3.5).
A linear multistep method is zero-stable if all roots of the characteristic equation that arises on applying the method to ′ = have magnitude less than or equal to unity, and that all roots with unit magnitude are simple. [2]
Explicit multistep methods can never be A-stable, just like explicit Runge–Kutta methods. Implicit multistep methods can only be A-stable if their order is at most 2. The latter result is known as the second Dahlquist barrier; it restricts the usefulness of linear multistep methods for stiff equations. An example of a second-order A-stable ...
For linear multistep methods, an additional concept called zero-stability is needed to explain the relation between local and global truncation errors. Linear multistep methods that satisfy the condition of zero-stability have the same relation between local and global errors as one-step methods.
Explicit examples from the linear multistep family include the Adams–Bashforth methods, and any Runge–Kutta method with a lower diagonal Butcher tableau is explicit. A loose rule of thumb dictates that stiff differential equations require the use of implicit schemes, whereas non-stiff problems can be solved more efficiently with explicit ...
1 Dahlquist Barriers. 1 comment. 2 AB coefficients. 1 comment. 3 It's correct. 4 Function of one variable or two? 1 comment. 5 Stability and A-stability. 2 comments.
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