Multi-Step Methods

Examples

Euler method
Reverse-Euler Method
Trapezoidal rule (ODEs)

One-step method

Euler method:

Order:

So order is

General multi-step methods

where .
Order of the method is the largest such that:

By Taylor expanding, we find


for .
Or rewriting:


This is necessary and sufficient condition.

By defining polynomials and we can write this concisely as

Dahlquist Equivalence Theorem

We say a polynomial obeys the root condition if all its roots are and all the roots of unit modulus are simple.
Then the multi-step method is convergent if and only if and obeys the root condition.

Constructing multi-step methods

Find a which obeys the root condition and .
Then find a s.t.

To see where this comes from, sub in and note that .