Chain Conditions in Commutative Rings
On this page:
Introduction
Given a method, we are particularly interested in the error that arises from the method itself and this is what the local truncation error measures.
Definition of Local Truncation Error
The IVPs we will consider are of the form
\[\begin{align*} y' = f(t, y), \quad y(a) = \alpha, \quad t \in [a, b] \end{align*}\]where $f$ is continuous and Lipschitz in $y$ on $[a, b] \times \RR$ for existence-uniqueness purposes. We define $h = (b - a)/n$, $t_j = a + jh$ for $j = 0, \ldots, n$, and $Y:[a, b] \to \RR$ to be the unique solution.
Single-Step
[definition] [deftitle]Definition (LTE of single-step method).[/deftitle]
Given a single-step method
\[\begin{align*} y_0 = \alpha; \qquad y_j = y_{j-1} + h\phi(t_{j-1}, y_{j-1}, y_j, h), \quad j \in [n], \end{align*}\]its local truncation error (abbv. LTE) is defined as
\[\begin{align*} \tau_j(h) = \frac{1}{h}\{Y(t_j) - [Y(t_{j-1}) + h\phi(t_{j-1}, Y(t_{j-1}, Y(t_j), h))]\} \end{align*}\]where $Y$ is the solution of the ODE. [/definition]
Linear Multistep
[definition] [deftitle]Definition (LTE of linear multistep method).[/deftitle]
Given an $m$-step linear method
\[\begin{align*} & y_0 = \alpha, \qquad y_i = \alpha_i, \quad i\in [m-1], \\ & \sum_{i=0}^m a_{m-i} y_{j-i+1} = h \sum_{i=0}^m b_{m-i} f_{j-i+1}, \end{align*}\]where $f_j \coloneqq f(t_j, y_j)$, its local truncation error (abbr. LTE) is defined as
\[\][/definition]