Second Dahlquist Barrier

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In the numerics of ordinary differential equations , the second Dahlquist barrier says that an A-stable linear multi-step method can have a maximum order of convergence of 2.

It was proven by Germund Dahlquist in 1963 . The proof also implies that the trapezoidal rule has the smallest error constant with 1/12 among all A-stable methods of order 2. This statement is sometimes given as part of the second Dahlquist barrier.

The barrier is a strong restriction on black box solvers for ordinary differential equations, since only methods with weaker stability properties are available for high order, which can fail for individual problems.

Alternatively, implicit Runge-Kutta methods can be used; according to the Daniel Moore conjecture , these can also be A-stable at almost any order of magnitude.

literature

  • Germund Dahlquist: A Special Stability Problem for Linear Multistep Methods in BIT 3 (1), 27-43, 1963
  • E. Hairer, G. Wanner: Solving Ordinary Differential Equations II, Stiff problems , Springer Verlag