Spruce up amr.rst

Docs/source/latex_theory/AMR/AMR.tex

@@ -22,7 +22,7 @@ \section{Mesh refinement}
 In addition, for some implementations where the field that is computed at a given level is affected by the solution at finer levels, there are cases where the procedure violates the integral of Gauss' Law around the refined patch, leading to long range errors \cite{Vaylpb2002,Colellajcp2010}. As will be shown below, in the procedure that has been developed in WarpX, the field at a given refinement level is not affected by the solution at finer levels, and is thus not affected by this type of error.
 
 \subsection{Electrostatic}
-A cornerstone of the Particle-In-Cell method is that assuming a particle lying in a hypothetical infinite grid, then if the grid is regular and symmetrical, and if the order of field gathering matches the order of charge (or current) deposition, then there is no self-force of the particle acting on itself: a) anywhere if using the so-called ``momentum conserving'' gathering scheme; b) on average within one cell if using the ``energy conserving'' gathering scheme \cite{Birdsalllangdon}. A breaking of the regularity and/or symmetry in the grid, whether it is from the use of irregular meshes or mesh refinement, and whether one uses finite difference, finite volume or finite elements, results in a net spurious self-force (which does not average to zero over one cell)  for a macroparticle close to the point of irregularity (mesh refinement interface for the current purpose) \cite{Vaylpb2002,Colellajcp2010}.
+A cornerstone of the Particle-In-Cell method is that, given a particle lying in a hypothetical infinite grid, if the grid is regular and symmetrical, and if the order of field gathering matches the order of charge (or current) deposition, then there is no self-force of the particle acting on itself: a) anywhere if using the so-called ``momentum conserving'' gathering scheme; b) on average within one cell if using the ``energy conserving'' gathering scheme \cite{Birdsalllangdon}. A breaking of the regularity and/or symmetry in the grid, whether it is from the use of irregular meshes or mesh refinement, and whether one uses finite difference, finite volume or finite elements, results in a net spurious self-force (which does not average to zero over one cell)  for a macroparticle close to the point of irregularity (mesh refinement interface for the current purpose) \cite{Vaylpb2002,Colellajcp2010}.
 
 A sketch of the implementation of mesh refinement in WarpX is given in Figure~\ref{fig:ESAMR} (left). Given the solution of the electric potential at a refinement level $L_n$, it is interpolated onto the boundaries of the grid patch(es) at the next refined level $L_{n+1}$. The electric potential is then computed at level $L_{n+1}$ by solving the Poisson equation. This procedure necessitates the knowledge of the charge density at every level of refinement. For efficiency, the macroparticle charge is deposited on the highest level patch that contains them, and the charge density of each patch is added recursively to lower levels, down to the lowest.
 

Docs/source/latex_theory/allbibs.bib

@@ -2181,7 +2181,6 @@ @article{LehePRE2016
 volume = {94},
 year = {2016}
 }
-
 @book{godfrey1985iprop,
 author = {Godfrey, B. B.},
 publisher = {Defense Technical Information Center},