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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]
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 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 ...
General linear methods (GLMs) are a large class of numerical methods used to obtain numerical solutions to ordinary differential equations. They include multistage Runge–Kutta methods that use intermediate collocation points , as well as linear multistep methods that save a finite time history of the solution.
Germund Dahlquist (16 January 1925 – 8 February 2005) was a Swedish mathematician known primarily for his early contributions to the theory of numerical analysis as applied to differential equations.
Dahlquist (1963) proposed the investigation of stability of numerical schemes when applied to nonlinear systems that satisfy a monotonicity condition. The corresponding concepts were defined as G-stability for multistep methods (and the related one-leg methods) and B-stability (Butcher, 1975) for Runge–Kutta
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.