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Accuracy

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

\begin{eqnarray*}
v_{n} &=& (L/T)\left(i_{n}-\left(i_{n}-T\left.\frac{di}{dt}\right\vert _{nT}
+ O(T^{2})\right)\right) \\
&=& \left. L\frac{di}{dt}\right\vert _{nT} + O(T)
\end{eqnarray*}

So the difference scheme approximates the continuous time equation to an accuracy that depends on $ T$ , the step size. Thus we expect that the discretization will do a better job as $ T$ gets small.

Performing the same analysis for the trapezoid rule yields:

\begin{eqnarray*}
v_{n} &=& L\left.\frac{di}{dt}\right\vert _{nT} + O(T^{2})
\end{eqnarray*}

So we say that the trapezoid rule is second-order accurate in $ T$ .


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``Discrete-Time Lumped Models'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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