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The (1+1)D Transmission Line

As a slightly more involved example, which highlights some of the issues which typically arise in the construction of these algorithms, consider the (1+1)D transmission line or telegrapher's equations [63]:
$\displaystyle \begin{eqnarray}l\frac{\partial i}{\partial t}+\frac{\partial u}{...
...c{\partial u}{\partial t}+\frac{\partial i}{\partial x}+gu+h&=&0 \end{eqnarray}$ (3.52a)

Here, $ i(x,t)$ and $ u(x,t)$ are the current and voltage in the transmission line, $ l$, $ c$, $ r$ and $ g$ are inductance, capacitance, resistance and shunt conductance per unit length respectively, and are all non-negative functions of $ x$ ($ l$ and $ c$ are strictly positive% latex2html id marker 80853
\setcounter{footnote}{2}\fnsymbol{footnote}). $ e(x,t)$ and $ h(x,t)$ represent distributed voltage and current source terms. System (3.56) is symmetric hyperbolic; it has the form of (3.1), with $ {\bf w} = [i,u]^{T}$, and

$\displaystyle {\bf P} = \begin{bmatrix}l&0\\ 0&c\\ \end{bmatrix}\hspace{0.3in}{...
...0&g\\ \end{bmatrix}\hspace{0.3in}{\bf f} = \begin{bmatrix}e\\ h\\ \end{bmatrix}$ (3.53)


Stefan Bilbao 2002-01-22