Averge scheme documented

Documentation of the average scheme for species properties.

doc/user-manual/bibliography.bib

@@ -51,4 +51,15 @@
   howpublished = {\url{https://gmsh.info/}},
 }
 
+@Article{welford1962note,
+  author    = {Welford, BP},
+  journal   = {Technometrics},
+  title     = {Note on a method for calculating corrected sums of squares and products},
+  year      = {1962},
+  number    = {3},
+  pages     = {419--420},
+  volume    = {4},
+  publisher = {Taylor \& Francis},
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}

doc/user-manual/fpakc_UserManual.tex

@@ -1,4 +1,4 @@
-\documentclass[10pt,a4paper,oneside]{book}
+\documentclass[10pt,a4paper,twoside]{book}
 \usepackage[latin1]{inputenc}
 \usepackage{amsmath}
 \usepackage{amsfonts}
@@ -22,6 +22,7 @@
     Author   = {Jorge Gonzalez}
   }
 }
+\usepackage[margin=1in]{geometry} % Reduces margins of the document
 % Allows breaking of URL in bibliography.
 \setcounter{biburllcpenalty}{7000}
 \setcounter{biburlucpenalty}{8000}
@@ -260,6 +261,18 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
     \section{Electromagnetic field}
       WIP.
+    
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    \section{Average scheme}
+      Particle-in-cell codes has an intrinsic statistical noise associated with them.
+      Although this can be reduced by increasing the number of particles, this also increases the CPU requirements of the case.
+      
+      It is quite common that most cases reach a quasi-steady state after a number of iterations and time-average results can be obtained after to improve analysis, plotting and restarting the case using these time-average results as new species backgrounds.
+      Although this is possible to do once the simulation is finished with post-processing tools, this is limited to the amount of iterations printed.
+      
+      \Gls{fpakc} implements a simple average scheme that, after a start time provided by the user, scores a mean and standard deviation of all the main species properties, and the electromagnetic field.
+      This scheme is based on the Welford's online algorithm~\cite{welford1962note}.
+      The averaged data is written in the same format as the input mesh at the end of the simulation.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
   \chapter{Installation}
@@ -642,7 +655,7 @@ make
           \item \textbf{initialTime}: Real.
                                       Units of $\unit{s}$.
                                       Initial simulation time.
-                                      If no value is provided, the initial time is set to $\unit[0]{s}$.
+                                      If no value is provided, the initial time is set to $\unit[0.0]{s}$.
           \item \textbf{pusher}: Character.
                                  Array dimension 'number of species'.
                                  Indicates the type of pusher used for each species:
@@ -684,6 +697,17 @@ make
                                                             File must be located at \textbf{output.path}.
                                   \end{itemize}
         \end{itemize}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+      \subsection{average}
+        This object determines the use of an average scheme.
+        If this object exists in the input file, average will be written at the end of the simulation.
+        Acceptable values are:
+        \begin{itemize}
+          \item \textbf{startTime}: Real.
+                                    Units in $\unit{s}$.
+                                    Simulation physical time in which average scheme will start to compute the mean and standard validation.
+                                    If no value is provided, the initial time is set to $\unit[0.0]{s}$.
+        \end{itemize}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
       \subsection{interactions}\label{ssec:input_interactions}
         This object determine the different interactions among species.