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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 ...
The coefficients are correct. I just checked the following paper: Purser, R. and L. Leslie. "Generalized Adams-Bashforth time integration schemes for a semi-Lagrangian model employing the second-derivative form of the horizontal momentum equations."
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.
The backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations.They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation.
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