## Definition

Difference-equation methods that involve only information from one previous mesh point, $$t_i$$ in the calculation for the mesh point, $$t_{i+1}$$ are called one-step methods. Since the approximate solution is available at each of the mesh points $$t_0 \dots t_i$$ before the approximation at $$t_{i+1}$$ is calculated, and because the error $$|\omega_i - y(t_i)|$$ tends to increase with $$j$$, it seems reasonable to develop methods that use these more accurate previous data when approximating the solution at $$t_{i+1}$$. Methods using the approximation at more than one previous mesh point to determine the approximation at the next point are called multistep methods.

An $$m$$-step multistep method for approximating the solution to the initial value problem:

$$\frac{dy}{dt} = f(t,y), \quad a \leq t \leq b, \quad y(a) = \alpha \\ \label{eq:mst1}$$

has a difference-equation for finding the approximation $$\omega_{i+1}$$ at the mesh point $$t_{i+1}$$ represented by the following equation, where $$m$$ is an integer greater than 1:

$$\omega_{i+1} = a_{m-1} \omega_i + a_{m-2} \omega_{i-1} + \dots + a_0 \omega_{i+1-m} + h[b_mf(t_{i+1}, \omega_{i+1}) + b_{m-1}f(t_i, \omega_i) + \dots + b_0f(t_{i+1-m}, \omega_{i+1-m})] \\ \label{eq:mst2}$$

for $$i=m-1 \dots N-1$$. The values $$a_{m-1} \dots a_0$$ and $$b_m \dots b_0$$ are constants and the starting values $$\omega_0 = \alpha_0 \dots \omega_{m-1} = \alpha_{m-1}$$ are specified. The starting values must be specified, generally by assuming $$\omega_0 = \alpha$$ and generating the remaining values by either a Runge-Kutta method or some other one-step technique. If $$b_m = 0$$ the method is called explicit, or open, since the multistep difference-equation then gives $$\omega_{i+1}$$ explicitly in terms of previously determined values. If $$b_m \ne 0$$ the method is called implicit, or closed, since $$\omega_{i+1}$$ occurs on both sides of the multistep difference-equation and is specified only implicitly. To apply an implicit method directly, the implicit equation must be solved for $$\omega_{i+1}$$.

The local truncation error for multistep methods is defined analogously to that of one-step methods. As in the case of one-step methods, the local truncation error provides a measure of how the solution to the differential equation fails to solve the difference equation. If $$\omega_{i+1}$$ is the $$(i+1)^{st}$$ step in a multistep method, the local truncation error at this step is:

$$\tau_{i+1}(h) = \frac{y(t_{i+1}) - a_{m-1}y(t_i) - a_0y(t_{i+1-m})}{h} - [b_mf(t_{i+1}, y(t_{i+1})) + \dots + b_0f(t_{i+1-m}, y(t_{i+1-m}))] \\ \label{eq:mst3}$$

for $$i=m-1 \dots N-1$$. In general, for an $$m$$-step explicit multistep method, the local truncation error is $$O(h^m)$$ and for an $$m$$-step implicit multistep method, the local truncation error is $$O(h^{m+1})$$.

## Common Multistep Methods

### Adams-Bashforth Two-Step Explicit Method

\begin{aligned} \omega_0 & = \alpha_0 \quad \omega_1 = \alpha_1 \\ \omega_{i+1} & = \omega_i + \frac{h}{2}[(3f(t_i, \omega_i) - f(t_{i-1}, \omega_{i-1})], \quad i=1,\dots,N-1 \\ \end{aligned} \label{eq:mst4}

The local truncation error is:

$$\tau_{i+1}(h) = h^2 \frac{5}{12} \left.\frac{d^3y}{dt^3}\right|_{t = \mu_i} \\ \label{eq:mst5}$$

For some $$\mu_i \in (t_{i-1}, t_{i+1})$$.

### Adams-Bashforth Three-Step Explicit Method

\begin{aligned} \omega_0 & = \alpha_0 \quad \omega_1 = \alpha_1 \quad \omega_2 = \alpha_2 \\ \omega_{i+1} & = \omega_i + \frac{h}{12}[(23f(t_i, \omega_i) - 16f(t_{i-1}, \omega_{i-1}) + 5f(t_{i-2}, \omega_{i-2})], \quad i=2,\dots,N-1 \\ \end{aligned} \label{eq:mst6}

The local truncation error is:

$$\tau_{i+1}(h) = h^3 \frac{3}{8} \left.\frac{d^4y}{dt^4}\right|_{t = \mu_i} \\ \label{eq:mst7}$$

For some $$\mu_i \in (t_{i-2}, t_{i+1})$$.

### Adams-Bashforth Four-Step Explicit Method

\begin{aligned} \omega_0 & = \alpha_0 \quad \omega_1 = \alpha_1 \quad \omega_2 = \alpha_2 \quad \omega_3 = \alpha_3 \\ \omega_{i+1} & = \omega_i + \frac{h}{24}[(55f(t_i, \omega_i) - 59f(t_{i-1}, \omega_{i-1}) + 37f(t_{i-2}, \omega_{i-2}) - 9f(t_{i-3}, \omega_{i-3}) ], \quad i=3,\dots,N-1 \\ \end{aligned} \label{eq:mst8}

The local truncation error is:

$$\tau_{i+1}(h) = h^4 \frac{251}{720} \left.\frac{d^5y}{dt^5}\right|_{t = \mu_i} \\ \label{eq:mst9}$$

For some $$\mu_i \in (t_{i-3}, t_{i+1})$$.

### Adams-Bashforth Five-Step Explicit Method

\begin{aligned} \omega_0 & = \alpha_0 \quad \omega_1 = \alpha_1 \quad \omega_2 = \alpha_2 \quad \omega_3 = \alpha_3 \quad \omega_4 = \alpha_4 \\ \omega_{i+1} & = \omega_i + \frac{h}{720}[(1901f(t_i, \omega_i) - 2774f(t_{i-1}, \omega_{i-1}) + 2616f(t_{i-2}, \omega_{i-2}) - 1274f(t_{i-3}, \omega_{i-3}) + 251f(t_{i-4}, \omega_{i-4})], \quad i=4,\dots,N-1 \\ \end{aligned} \label{eq:mst10}

The local truncation error is:

$$\tau_{i+1}(h) = h^5 \frac{95}{288} \left.\frac{d^6y}{dt^6}\right|_{t = \mu_i} \\ \label{eq:mst11}$$

For some $$\mu_i \in (t_{i-4}, t_{i+1})$$.

### Adams-Moulton Two-Step Implicit Method

\begin{aligned} \omega_0 & = \alpha_0 \quad \omega_1 = \alpha_1 \\ \omega_{i+1} & = \omega_i + \frac{h}{12}[(5f(t_{i+1}, \omega_{i+1}) + 8f(t_{i}, \omega_{i}) - f(t_{i-1}, \omega_{i-1})], \quad i=1,\dots,N-1 \\ \end{aligned} \label{eq:mst12}

The local truncation error is:

$$\tau_{i+1}(h) = -h^3 \frac{1}{24} \left.\frac{d^4y}{dt^4}\right|_{t = \mu_i} \\ \label{eq:mst13}$$

For some $$\mu_i \in (t_{i-1}, t_{i+1})$$.

### Adams-Moulton Three-Step Implicit Method

\begin{aligned} \omega_0 & = \alpha_0 \quad \omega_1 = \alpha_1 \quad \omega_2 = \alpha_2 \\ \omega_{i+1} & = \omega_i + \frac{h}{24}[(9f(t_{i+1}, \omega_{i+1}) + 19f(t_{i}, \omega_{i}) - 5f(t_{i-1}, \omega_{i-1}) + f(t_{i-2}, \omega_{i-2})], \quad i=2,\dots,N-1 \\ \end{aligned} \label{eq:mst14}

The local truncation error is:

$$\tau_{i+1}(h) = -h^4 \frac{19}{720} \left.\frac{d^5y}{dt^5}\right|_{t = \mu_i} \\ \label{eq:mst15}$$

For some $$\mu_i \in (t_{i-2}, t_{i+1})$$.

### Adams-Moulton Four-Step Implicit Method

\begin{aligned} \omega_0 & = \alpha_0 \quad \omega_1 = \alpha_1 \quad \omega_2 = \alpha_2 \quad \omega_3= \alpha_3 \\ \omega_{i+1} & = \omega_i + \frac{h}{720}[(251f(t_{i+1}, \omega_{i+1}) + 646f(t_{i}, \omega_{i}) - 264f(t_{i-1}, \omega_{i-1}) + 106f(t_{i-2}, \omega_{i-2}) - 19f(t_{i-3}, \omega_{i-3})], \quad i=3,\dots,N-1 \\ \end{aligned} \label{eq:mst16}

The local truncation error is:

$$\tau_{i+1}(h) = -h^5 \frac{3}{160} \left.\frac{d^6y}{dt^6}\right|_{t = \mu_i} \\ \label{eq:mst17}$$

For some $$\mu_i \in (t_{i-3}, t_{i+1})$$.

## Summary

It is interesting to compare an m-step Adams-Bashforth explicit method to a $$(m-1)$$ step Adams-Moulton implicit method. Both involve $$m$$ evaluations of $$f(t,y)$$ per step, and both have the terms $$O(h^m)$$ in their local truncation errors. In general, the coefficients of the terms involving $$f(t,y)$$ in the local truncation error are smaller for the implicit methods than for the explicit methods. This leads to greater stability and smaller round off errors for the implicit methods.