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\Large{\bf I. INTRODUCTION}
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\noindent \rm This document describes the equations used in TITAN, a one-dimensional adaptive-grid radiation hydrodynamics code intended for astrophysical calculations. Suggestions about how to use the code are given in the TITAN Users Guide. TITAN follows the basic philosophy of WH80s, the very powerful code written by Karl-Heinz Winkler, except we use the grid equation developed by Dorfi and Drury which is somewhat simpler to implement and use by average users. Any potential user of this code should read carefully the following publications describing WH80s:

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\rm ({\bf 1}) WH80s: Numerical Radiation Hydrodynamics, K.--H. Winkler and \\
\hspace{10mm}M.L. Norman,\ in \it Astrophysical Radiation Hydrodynamics,\\
\hspace{10mm}\ \rm (Dordrecht: Reidel), pp. 71--139, 1986\\

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\rm ({\bf 2}) Implicit Adaptive--Grid Radiation Hydrodynamics, K.--H. Winkler\\
\hspace{10mm}M.L. Norman, and D. Mihalas in \it Multiple Time Scales,\\
\hspace{10mm}\rm Computational Techniques, Vol. \bf 2, \rm ed. J.U. Brackbill\\
\hspace{10mm}\rm and B.I. Cohen, (New York: Academic Press), pp. 145 -- 184, 1985\\

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\rm ({\bf 3}) Adaptive--Mesh Radiation Hydrodynamics. I. The Radiation Transport\\
\hspace{10mm}Equation in a Completely Adaptive Coordinate System,\\
\hspace{10mm}K.--H. Winkler, M.L. Norman, and D. Mihalas, \it J.Q.S.R.T.,\\
\hspace{10mm}\bf 31, \rm 473, 1984\\

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\rm ({\bf 4}) Adaptive--Mesh Radiation Hydrodynamics. II. The Radiation and\\
\hspace{10mm}Fluid Equations in Relativistic Flows, D. Mihalas, K.--H. Winkler,\\
\hspace{10mm}and M.L. Norman, \it J.Q.S.R.T., \bf 31, \rm 479, 1984\\
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\noindent \rm In what follows, each of the seven basic equations are discussed in separate sections. In general there are five conservation relations (continuity, gas momentum, radiating fluid energy, radiation energy, and radiation momentum), a mass definition equation, and an adaptive grid equation. The code is written so that it can treat ordinary hydrodynamics without radiation, full radiation hydrodynamics, and time-dependent radiation in a static medium. Further, users can specify an Eulerian grid, a Lagrangean grid, or an adaptive grid. At the beginning of a section, the subsection titled "differential equation" actually contains the original differential equation integrated over a finite volume, and transformed to adaptive coordinates, as described in the references ({\bf 1}) and ({\bf 3}) above. The advantage of this approach is that the difference equations are strictly conservative (apart from undifferentiated and source/sink terms).\\

\noindent \rm In finite differencing the equations we use a staggered mesh. The medium is assumed to consist of $N$ cells, bounded by $N + 1$ interfaces. The interface $k = 1$ is the leftmost boundary of the domain, and the interface $k = N + 1$ is the rightmost boundary of the domain. In stratified media (e.g. a star) $k = 1$ is the innermost boundary and $k = N + 1$ is the outermost boundary.
All thermodynamic variables are cell-centered; thus the gas density $\rho_{k}$, temperature $T_{k}$, gas internal energy $e_{k}$, gas pressure $p_{k}$, and radiation pressure $P_{k}$ are the values of these variables at the center of the cell $(k, k+1)$. Likewise the radius $r_{k}$, mass $m_{k}$, velocity $u_{k}$, and radiation flux $F_{k}$ are all centered on interface $k$.
The full nonlinear difference equations are then linearized. The linearized system for the corrections has a block pentadiagonal form. The solution of the system is obtained by the standard Newton-Raphson iteration procedure.\\

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\noindent \rm To assist the reader we write all ordinary physical and mathematical variables in {\it italic type}, and {\tt FORTRAN} quantities in {\tt typewriter type}. In naming {\tt FORTRAN} variables, we also use the elegant mnemonic
system devised by Winkler. For example, we denote the matrices $E_{-2}$, $E_{-1}$, $E_{0}$, $E_{1}$, and $E_{2}$ of the linearized system as {\tt em2, em1, e00, ep1, and ep2}. For brevity, because several physical terms in one of the difference equations may contribute to a given matrix element, we use the notation

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{\tt enn(i,j):}\ function(physical variables)\\
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\noindent to denote that the quantity on the right--hand side is {\it added into} the specified matrix element.

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