1.3 Order of Accuracy

1.3.4 Definition of Multi-Step Methods

Measurable Outcome 1.5, Measurable Outcome 1.8, Measurable Outcome 1.13, Measurable Outcome 1.15

The class of finite difference methods known as multi-step methods is one of the most widely-used approaches for solving ordinary differential equations, and forms the basis for solving partial differential equations as well.

Definition of Multi-Step Methods

The generic form of an \(s\)-step multi-step method is,

\[v^{n+1} + \sum _{i=1}^ s \alpha _ i v^{n+1-i} = {\Delta t}\sum _{i=0}^ s \beta _ i f^{n+1-i}.\]

(1.51)

A multi-step method with \(\beta _0 = 0\) is known as an explicit method since in this case the new value \(v^{n+1}\) can be determined as an explicit function of known values (i.e. from \(v^ i\) and \(f_ i\) with \(i \leq n\)). A multi-step method with \(\beta _0 \neq 0\) is known as an implicit method since in this case the new value \(v^{n+1}\) is an implicit function of itself through the forcing function, \(f^{n+1} = f(v^{n+1},t^{n+1})\). We defer implicit methods to Section 1.7.1 and focus on explicit methods here.