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Accuracy of Backward Euler

Suppose we take the backward-difference approximation $ f_{n}=(m/T)(v_{n}-v_{n-1})$ , and expand $ v_{n-1}$ in Taylor series about $ v_{n}$ . This yields:

\begin{eqnarray*}
f_{n} &=& \frac{m}{T}\left(v_{n}-\left(v_{n}-T\left.\frac{dv}{dt}\right\vert _{nT}
+ {\cal O}(T^{2})\right)\right) \\ [10pt]
&=& \left. m\frac{dv}{dt}\right\vert _{nT} + {\cal O}(T)
\end{eqnarray*}


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``Introduction to Physical Signal Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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