1.5 Zero Stability and the Dahlquist Equivalence Theorem

1.5.3 Dahlquist Equivalence Theorem

Measurable Outcome 1.7, Measurable Outcome 1.8, Measurable Outcome 1.9, Measurable Outcome 1.10

In order for a multi-step method to be convergent (as described in Section 1.4.2), two conditions must be met:

Consistency: In the limit of \({\Delta t}\rightarrow 0\), the method must be a consistent discretization of the ordinary differential equation.

Stability In the limit of \({\Delta t}\rightarrow 0\), the method must not have solutions that can grow unbounded as \(n = T/{\Delta t}\rightarrow \infty\).

The Dahlquist Equivalence Theorem in fact guarantees that a consistent and stable multi-step method is convergent, and vice-versa:

Dahlquist Equivalence Theorem

A multi-step method is convergent if and only if it is consistent and stable.

Exercise The numerical scheme defined by the equation \(v^{n+1} + 4v^ n -5v^{n-1} = 4\Delta t f^ n + 2\Delta t f^{n-1}\) is

Exercise 1

Answer: The method is consistent, but not zero stable. Since it is not zero stable, the Dahlquist equivalence theorem tells us that the method is not convergent.