# 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.

Exercise 1 What are the non-zero \(\alpha _ i\) and \(\beta _ i\) for the forward Euler method?

Exercise 1
\(\alpha _1=-1, \beta _1=1\)
\(\alpha _1=1, \beta _1=2\)
\(\alpha _2=-1, \beta _1=2\)
\(\alpha _1=-1, \beta _1=2\)

Answer:

Using the notation given above, the forward Euler method is:

\[\alpha _1 = -1 \quad \mbox{all other} \quad \alpha _ i = 0\] | (1.52)

\[\beta _1 = 1 \quad \mbox{all other} \quad \beta _ i = 0\] | (1.53)

Exercise 2 What are the non-zero \(\alpha _ i\) and \(\beta _ i\) for the Midpoint method?

Exercise 2
\(\alpha _1=-1, \beta _1=1\)
\(\alpha _1=1, \beta _1=2\)
\(\alpha _2=-1, \beta _1=2\)
\(\alpha _1=-1, \beta _1=2\)

Answer:

Using the notation given above, the midpoint method is:

\[\alpha _2 = -1 \quad \mbox{all other} \quad \alpha _ i = 0\] | (1.54)

\[\beta _1 = 2 \quad \mbox{all other} \quad \beta _ i = 0\] | (1.55)