3-NIRVANA-user-guide/3.1-Code-basics.md

@@ -389,9 +389,9 @@ The mesh refinement algorithm relies on the oct-tree data structure of
 generic blocks (fixed size of 4 cells per direction) represented by the
 master mesh pointer `_G0` (of type `*GRD`). A generic block of
 refinement level $l$ has cell spacings
-$(\d x^{(l)}, \d y^{(l)}, \d z^{(l)})=
-(\d x^{(0)},\d y^{(0)},\d z^{(0)})/2^l$ where
-$(\d x^{(0)},\d y^{(0)},\d z^{(0)})$ is the cell spacing of the base
+$(\delta x^{(l)}, \delta y^{(l)}, \delta z^{(l)})=
+(\delta x^{(0)},\delta y^{(0)},\delta z^{(0)})/2^l$ where
+$(\delta x^{(0)},\delta y^{(0)},\delta z^{(0)})$ is the cell spacing of the base
 level, i.e., resolution doubles each time when refined.
 
 Starting on the base level ($l=0$) mesh refinements are realized by
@@ -405,24 +405,24 @@ individually selected by the user. The standard implementation covers
 the following criteria:
 
 -   derivatives-based (most important in practice):
-    $\left[\alpha\frac{|\d U|}{|U|+U_\mathrm{ref}}+(1-\alpha)
-    \frac{|\d^2U|}{|\d U|+{\rm FILTER}\cdot(|U|+U_\mathrm{ref})}\right]
-    \left(\frac{\d x^{(l)}}{\d x^{(0)}}\right)^{\xi}
+    $$\left[\alpha\frac{|\delta U|}{|U|+U_\mathrm{ref}}+(1-\alpha)
+    \frac{|\delta^2U|}{|\delta U|+{\rm FILTER}\cdot(|U|+U_\mathrm{ref})}\right]
+    \left(\frac{\delta x^{(l)}}{\delta x^{(0)}}\right)^{\xi}
     \left\{\begin{array}{ll}
     >\mathcal{E}_{U}  &\exists U \quad\hbox{refinement}\\
     <0.8\mathcal{E}_{U}  & \forall U \quad\hbox{derefinement}\\
-    \end{array}\right.$
+    \end{array}\right.$$
 
     The criterion is applied to physical variables $U$
     ($=\{\varrho,\mathbf{m},e,\mathbf{B},C_\mathrm{c}\}$). Undivided
-    first ($\d U$) and second ($\d^2U$) differences are computed along
+    first ($\delta U$) and second ($\delta^2U$) differences are computed along
     various spatial directions. The switch $\alpha\in [0,1]$ (code
     parameter: `_C.amr_d1`) selects between a purely gradient-based
     criterion ($\alpha =1$) and a purely second-derivatives-based
     criterion ($\alpha =0$). $U_\mathrm{ref}$ are reference values to
     avoid division by zero for indefinite variables. The exponent
     $\xi\in [0,2]$ (code parameter: `_C.amr_exp`) introduces a power law
-    functional on the grid spacing ratio $\d x^{(l)}/\d x^{(0)}$ which
+    functional on the grid spacing ratio $\delta x^{(l)}/\delta x^{(0)}$ which
     allows to tune the behavior with progressive mesh refinement. The
     macro `FILTER` (preset to $10^{-2}$) is a filter to suppress
     refinement at small-scale solution wiggles. The individual
@@ -439,11 +439,11 @@ the following criteria:
 -   Jeans-length-based (important in simulations involving selfgravity):
     $$\left(\frac{\pi}{G}\frac{c_s^2}{\varrho}\right)^{1/2}
     \cdot \mathcal{E}_\mathrm{Jeans}\left\{\begin{array}{ll}
-    <\d s & \mbox{refinement}\\
-    >1.25\d s & \mbox{derefinement}\\
+    <\delta s & \mbox{refinement}\\
+    >1.25\delta s & \mbox{derefinement}\\
     \end{array}\right.$$ where the first factor is the local Jeans
     length with $c_s$ the sound speed and $G$ the gravitational
-    constant, and where $\d s=\min\{\d x,h_y\d y, h_z\d z\}$.
+    constant, and where $\delta s=\min\{\delta x,h_y\delta y, h_z\delta z\}$.
     $\mathcal{E}_\mathrm{Jeans}$ is a user-specific threshold giving the
     fraction of the local Jeans length to be resolved by at least 1 grid
     cell.
@@ -453,8 +453,8 @@ the following criteria:
     $$2\pi \left(\frac{\kappa T}{\max\left\{T\left(\frac{\partial L}{\partial T}\right)_{\varrho}
     -\varrho \left(\frac{\partial L}{\partial \varrho}\right)_{T},EPS\right\}}
     \right)^{1/2}\cdot \mathcal{E}_\mathrm{Field}\left\{\begin{array}{ll}
-    <\d s & \mbox{refinement}\\
-    >1.25\d s & \mbox{derefinement}\\
+    <\delta s & \mbox{refinement}\\
+    >1.25\delta s & \mbox{derefinement}\\
     \end{array}\right.$$ where the first factor is the Field length with
     $L(\varrho, T)$ the density- and temperature-dependent heatloss
     function and $\kappa$ the thermal conduction coefficient.