Beginning of some documentation.

git-svn-id: https://svn.cct.lsu.edu/repos/numrel/LSUThorns/SummationByParts/trunk@24 f69c4107-0314-4c4f-9ad4-17e986b73f4a

1 files changed, 88 insertions, 4 deletions

diff --git a/doc/documentation.tex b/doc/documentation.tex
index 65e0655..676418c 100644
--- a/doc/documentation.tex
+++ b/doc/documentation.tex
@@ -98,7 +98,7 @@
 
 % Add an abstract for this thorn's documentation
 \begin{abstract}
-Calculate derivates of grid functions using finite difference stencils
+Calculate first derivates of grid functions using finite difference stencils
 that satisfy summation by parts.
 \end{abstract}
 
@@ -106,10 +106,92 @@ that satisfy summation by parts.
 % Remove them or add your own.
 
 \section{Introduction}
-
-\section{Physical System}
-
+Given a discretization $x_0\ldots x_N$ of a computational domain $x\in[a,b]$
+with gridspacing $h$ a one dimensional finite difference operator
+approximation to a first derivative, $D$, is said to satisfy summation by
+parts (SBP) with respect to a scalar product (defined by its
+coefficients $\sigma_{ij}$)
+\begin{equation}
+\langle u, v\rangle_h = h \sum_{i=0}^{N} u_i v_j \sigma_{ij}
+\end{equation}
+if the property
+\begin{equation}
+\langle u, Dv\rangle_h +\langle Du, v\rangle_h = \left . (uv)\right|^b_a
+\end{equation}
+is satisfied for all possible gridfunctions $u$ and $v$. 
+
+At a given finite difference order, there are several different ways of doing
+this depending on the structure of the scalar product. The three commonly
+considered cases are the diagonal norm, the restricted full norm and the full
+norm (see figure~\ref{fig:norm} for the structure).
+\begin{figure}[t]
+\[
+\sigma = \left ( \begin{array}{cccccc}
+           x &   &   &   &   & \\
+             & x &   &   &   & \\
+             &   & x &   &   & \\
+             &   &   & x &   & \\
+             &   &   &   & 1 & \\
+             &   &   &   &   & \ddots
+         \end{array} \right ),\mbox{\hspace{0.2em}}
+\sigma = \left ( \begin{array}{ccccccc}
+           x &   &   &   &   &   & \\
+             & x & x & x & x &   & \\
+             & x & x & x & x &   & \\
+             & x & x & x & x &   & \\
+             & x & x & x & x &   & \\
+             &   &   &   &   & 1 & \\
+             &   &   &   &   &   & \ddots
+        \end{array} \right ),\mbox{\hspace{0.2em}}
+\sigma = \left ( \begin{array}{cccccc}
+           x & x & x & x &   & \\
+           x & x & x & x &   & \\
+           x & x & x & x &   & \\
+           x & x & x & x &   & \\
+             &   &   &   & 1 & \\
+             &   &   &   &   & \ddots
+        \end{array} \right )
+\]
+\caption{The structure of the scalar product matrix in the diagonal case
+(left), the restricted full case (middle) and the full case (right) for the
+4th order interior operators. Only non-zero elements are shown.}
+\label{fig:norm}
+\end{figure}
+
+In the following we denote the order of accuracy at the boundary by $\tau$, 
+the order in the interior by $s$ and the width of the boundary 
+region\footnote{The width of the region where standard centered finite
+differences are not used.} by $r$.
+
+For the diagonal norm case it turns out that with $r=2\tau$ it is possible
+to find SBP operators with $s=2\tau$ (at least when $\tau\le 4$), i.e.\ the
+order of accuracy at the boundary is half the order in the interior. For the
+restricted full norm case $\tau = s-1$ when $r=\tau +2$ and for the full norm
+case $\tau = s-1$ when $r=\tau + 1$.
+
+The operators are named after their norm and their interior and boundary
+orders. Thus, for example, we talk about diagonal norm 6-3 operators and
+restricted full norm 4-3 operators.
+
+In the diagonal case the 2-1 and 4-3 operators are unique whereas the 6-3
+and 8-4 operators have 1 and 3 free parameters, respectively. In the restricted
+full norm case the 4-3 operators have 3 free parameters and the 6-5 operators
+have 4, while in the full norm case the number of free parameters are 1 less
+than the restricted full case.
 \section{Numerical Implementation}
+Currently this thorn implements only diagonal and restricted full norm SBP
+operators. The diagonal norm 2-1, 4-2 and 6-3 and the restricted full norm
+4-3 are the ones listed in \cite{strand93} where in the presence of free
+parameters the set of parameters giving a minimal bandwidth have been chosen.
+For the diagonal norm 8-4 case the minimal bandwidth choice are to restrictive
+with respect to the Courant factor and in this case parameters are chosen
+so as to maximize the Courant factor\footnote{This is done by choosing
+parameters that minimizes the largest eigenvalue of the amplification matrix
+for a simple 1D advection equation with a penalty method implementation of
+periodic boundary conditions as suggested in \cite{lrt05}.}. In addition the
+restricted full norm 6-5 operator has been implemented. This was calculated
+using a Mathematica script kindly provided by Jos\'{e} M.
+Mart\'{\i}n-Garc\'{\i}a.
 
 \section{Using This Thorn}
 
@@ -135,6 +217,8 @@ that satisfy summation by parts.
 
 
 \begin{thebibliography}{9}
+\bibitem{strand93} Bo Strand, 1994, Journal of Computational Physics, 110, 47-67
+\bibitem{lrt05} Luis Lehner, Oscar Reula and Manuel Tiglio, in preparation.
 
 \end{thebibliography}