Added theroy blurb about the exact force calculation for the gravity scheme.

theory/Multipoles/bibliography.bib

@@ -191,4 +191,61 @@ keywords = "adaptive algorithms"
   adsnote = {Provided by the SAO/NASA Astrophysics Data System}
 }
 
+@ARTICLE{Hubber2011,
+   author = {{Hubber}, D.~A. and {Batty}, C.~P. and {McLeod}, A. and {Whitworth}, A.~P.
+	},
+    title = "{SEREN - a new SPH code for star and planet formation simulations. Algorithms and tests}",
+  journal = {\aap},
+ keywords = {hydrodynamics, methods: numerical, stars: formation},
+     year = 2011,
+    month = may,
+   volume = 529,
+      eid = {A27},
+    pages = {A27},
+      doi = {10.1051/0004-6361/201014949},
+   adsurl = {http://adsabs.harvard.edu/abs/2011A%26A...529A..27H},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+
+@ARTICLE{Klessen1997,
+   author = {{Klessen}, R.},
+    title = "{GRAPESPH with fully periodic boundary conditions - Fragmentation of molecular clouds}",
+  journal = {\mnras},
+ keywords = {Molecular Clouds, Interstellar Matter, Fragmentation, Astronomical Models, Computer Programs, Boundary Conditions},
+     year = 1997,
+    month = nov,
+   volume = 292,
+    pages = {11},
+      doi = {10.1093/mnras/292.1.11},
+   adsurl = {http://adsabs.harvard.edu/abs/1997MNRAS.292...11K},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+@ARTICLE{Hernquist1991,
+   author = {{Hernquist}, L. and {Bouchet}, F.~R. and {Suto}, Y.},
+    title = "{Application of the Ewald method to cosmological N-body simulations}",
+  journal = {\apjs},
+ keywords = {Computational Astrophysics, Galactic Structure, Hubble Constant, Many Body Problem, Astronomical Models, Boundary Conditions, Spatial Resolution},
+     year = 1991,
+    month = feb,
+   volume = 75,
+    pages = {231-240},
+      doi = {10.1086/191530},
+   adsurl = {http://adsabs.harvard.edu/abs/1991ApJS...75..231H},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+@ARTICLE{Ewald1921,
+   author = {{Ewald}, P.~P.},
+    title = "{Die Berechnung optischer und elektrostatischer Gitterpotentiale}",
+  journal = {Annalen der Physik},
+     year = 1921,
+   volume = 369,
+    pages = {253-287},
+      doi = {10.1002/andp.19213690304},
+   adsurl = {http://adsabs.harvard.edu/abs/1921AnP...369..253E},
+  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
 

theory/Multipoles/exact_forces.tex

+\subsection{Exact forces for accuracy checks}
+\label{ssec:exact_forces}
+
+To assess the accuracy of the gravity solver, \swift can also compute
+the gravitational forces and potential for a subset of particles using
+a simple direct summation method. This is obviously much slower and
+should only be used for code testing purposes. The forces for a
+selection of particles are computed every time-step if they are active
+and dumped to a file alongside the forces computed by the FMM method.
+
+In the case where periodic boundary conditions are used, we apply the
+\cite{Ewald1921} summation technique to include the contribution to
+the forces of all the infinite periodic replications of the particle
+distribution. We use the approximation to the infinite series of terms
+proposed by \cite{Hernquist1991}\footnote{Note, however, that there is
+a typo in their formula for the force correction terms. The correct
+expression is given by \cite{Klessen1997} \citep[see
+also][]{Hubber2011}.}, which we tabulate in one octant using 64
+equally spaced bins along each spatial direction spanning and the
+range $[0,L]$, where $L$ is the side-length of the box.

theory/Multipoles/fmm_standalone.tex

@@ -35,6 +35,7 @@ Making gravity great again.
 \input{fmm_summary}
 %\input{gravity_derivatives}
 \input{mesh_summary}
+\input{exact_forces}
 
 \bibliographystyle{mnras}
 \bibliography{./bibliography.bib}