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\title{
  The {\tt RNPL} \ User's Guide
}
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\author{Robert Marsa \\ 
        Matthew Choptuik \\ 
        Center for Relativity \\ 
        The University of Texas at Austin \\ 
      Austin, TX, 78712-1081 \\ [0.10cm]
        marsa@hoffmann.ph.utexas.edu \\ 
        matt@infeld.ph.utexas.edu \\ 
}
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\date{May 1995}
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\begin{document}
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{\newpage
\clearpage
\samepage \begin{eqnarray*}\partial_t^2 \phi = \partial_x^2 \phi + \partial_y^2 \phi + \partial_z^2
\phi \\ 
\phi(xmin,y,z,t) = 0 \\ 
\phi(xmax,y,z,t) = 0 \\ 
\phi(x,ymin,z,t) = 0 \\ 
\phi(x,ymax,z,t) = 0 \\ 
\phi(x,y,zmin,t) = 0 \\ 
\phi(x,y,zmax,t) = 0 \\ 
\phi(x,y,z,0) = A e^{{\frac{x-c_x}{\delta_x}}^2}
e^{{\frac{y-c_y}{\delta_y}}^2} e^{{\frac{z-c_z}{\delta_z}}^2} \\ 
where \quad xmin < x < xmax, ymin < y < ymax, zmin < z < zmax 
\end{eqnarray*}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{eqnarray*}\partial_t^2\phi(x,t) = (1-\beta^2)\partial_x^2\phi(x,t) +
2\beta\partial_t\partial_x\phi(x,t) \\ 
\phi(xmin,t) = 0 \\ 
\phi(xmax,t) = 0 \\ 
\phi(x,0) = A e^{\frac{-(x-c)^2}{\Delta^2}} \\ 
\partial_t\phi(x,0) = \frac{-2(x-c)}{\Delta^2} A
e^{\frac{-(x-c)^2}{\Delta^2}} \\ 
\beta(x) = 0.5 \\ 
where \quad xmin < x < xmax 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\Phi \equiv \partial_x \phi \\ 
\Pi \equiv \partial_t \phi - \beta \partial_x \phi 
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\partial_t\Phi(x,t) = \partial_x(\beta \Phi + \Pi) \\ 
\partial_t\Pi(x,t) = \partial_x(\beta \Pi + \Phi) \\ 
\Phi(x,0) = \frac{-2(x-c)}{\Delta^2} A e^{\frac{-(x-c)^2}{\Delta^2}} \\ 
\Pi(x,0) = \Phi(x,0) \\ 
\partial_x\Phi(xmin,t) = 0 \\ 
\partial_x\Phi(xmax,t) = 0 \\ 
\Pi(xmin,t) = 0 \\ 
\Pi(xmax,t) = 0 \\ 
\end{eqnarray*}
}

\stepcounter{chapter}
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{\newpage
\clearpage
\samepage \begin{eqnarray*}\partial_t^2\phi(x,t) = \partial_x^2\phi(x,t) \\ 
\phi(x,0) = A e^{\frac{-(x-c)^2}{\Delta^2}} \\ 
\partial_t\phi(x,0) = \frac{-2(x-c)}{\Delta^2} A
e^{\frac{-(x-c)^2}{\Delta^2}} \\ 
\phi(xmin,t) = \phi(xmax,t) \\ 
where \quad xmin < x < xmax 
\end{eqnarray*}
}

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