29.1. Foundations#
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29.1.1. Local truncation error and Consistency#
Let
\[y_{n+1} = \Psi(t_{n+1}; y_n, ..., y_{n-k+1}; h, f(\cdot)) = y_n + h A(t_{n+1}; y_n, ..., y_{n-k+1}; h, f(\cdot)) \ ,\]
the numerical method for the solution of the initial value problem
\[\begin{split}\begin{aligned}
\dot{y}(t) & = f(y(t),t) \\
y(0) & = y_0 \ .
\end{aligned}\end{split}\]
The local truncation error is the error produced by one step of the method, i.e.
\[\delta_{n+k}^h := y_{n+k} - y(t_{n+k}) \ ,\]
being \(y_{n+k}\) the result of one step of the numerical method, and \(y(t_{n+k})\) the exact solution using the same initial conditions (assuming, there’s no error in previous steps). A numerical method is defined consistent if
\[\lim_{h \rightarrow 0} \frac{\delta^h_{n+k}}{h} = 0 \ .\]
The method has order \(p\) if
\[\delta^h_{n+k} = O(h^{p+1}) \quad , \quad \text{for } h \rightarrow 0 \ .\]
29.1.2. Global truncation error and Convergence#
The global truncation error \(e_n\) is the accumulation of the error up to time \(t_n\),
\[e_n^h := y_{n} - y(t_n) \ .\]
A numerical method is defined convergent if
\[\lim_{h \rightarrow 0} \max_n | e_{n} | = 0 \ .\]