Corrections to the kernel definition TeX document

theory/kernel/kernel_definitions.tex

@@ -36,16 +36,16 @@ hence be written (in 3D) as
  W(\vec{x},h) \equiv \frac{1}{H^3}f\left(\frac{|\vec{x}|}{H}\right),
 \end{equation}
 
-where $H(h)$ is defined below and $f(u)$ is a dimensionless function,
-usually a low-order polynomial, such that $f(u \geq 1) = 0$ and
-normalised such that
+where $H=\gamma h$ is defined below and $f(u)$ is a dimensionless
+function, usually a low-order polynomial, such that $f(u \geq 1) = 0$
+and normalised such that
 
 \begin{equation}
   \int f(|\vec{u}|){\rm d}^3u = 1.
 \end{equation}
 
 $H$ is the kernel's support radius and is used as the ``smoothing
-length'' in the Gadget code( {i.e.} $H(h)=h$). This definition is,
+length'' in the Gadget code( {i.e.} $H=h$). This definition is,
 however, not very physical and makes comparison of kernels at a
 \emph{fixed resolution} difficult. A more sensible definition of the
 smoothing length, related to the Taylor expansion of the
@@ -64,8 +64,8 @@ The smoothing length is then:
 \end{equation}
 
 Each kernel, {\it i.e.} defintion of $f(u)$, will have a different
-ratio $h/H$. So for a \emph{fixed resolution} $h$, one will have
-different kernel support sizes, $H$, and different number of
+ratio $\gamma = H/h$. So for a \emph{fixed resolution} $h$, one will
+have different kernel support sizes, $H$, and a different number of
 neighbours, $N_{\rm ngb}$ to interact with. One would typically choose
 $h$ for a simulation as a multiple $\eta$ of the mean-interparticle
 separation:
@@ -87,9 +87,10 @@ useful quantity to use in implementations of SPH. It is defined as (in
 
 Once the fixed ratio $\gamma= H/h$ is known (via equations
 \ref{eq:sph:sigma} and \ref{eq:sph:h}) for a given kernel, the number
-of neighbours only depends on the resolution $\eta$.  For the usual
-cubic spline kernel (see below), setting the simulation resolution to
-$\eta=1.2348$ yields the commonly used value $N_{\rm ngb} = 48$.
+of neighbours only depends on the resolution parameter $\eta$.  For
+the usual cubic spline kernel (see below), setting the simulation
+resolution to $\eta=1.2348$ yields the commonly used value $N_{\rm
+  ngb} = 48$.
 
 \section{Kernels available in \swift}
 
@@ -130,35 +131,21 @@ The kernel function $f(u)$ reads:
 \end{equation}
 
 
-\subsubsection{Quartic spline ($M_5$) kernel}
-
-In 3D, we have $C=\frac{15625}{512\pi}$ and $\gamma=H/h = 1.825742$.\\
-The kernel function $f(u)$ reads:
-
-\begin{equation}
-  M_4(u) = \left\lbrace\begin{array}{rcl}
-  3u^3 - 3u^2 + \frac{1}{2} & \mbox{if} & u<\frac{1}{2}\\
-  -u^3 + 3u^2 - 3u + 1 & \mbox{if} & u \geq \frac{1}{2}
-  \end{array}
-  \right.
-    \nonumber
-\end{equation}
-
-
 \subsubsection{Quartic spline ($M_5$) kernel}
 
 In 3D, we have $C=\frac{15625}{512\pi}$ and $\gamma=H/h = 2.018932$.\\
 The kernel function $f(u)$ reads:
 
-\begin{equation}
-  M_5(u) = \left\lbrace\begin{array}{rcl}
+\begin{align}
+  M_5(u) &=     \nonumber\\
+  &\left\lbrace\begin{array}{rcl}
   6u^4 - \frac{12}{5}u^2 + \frac{46}{125} & \mbox{if} & u < \frac{1}{5} \\
   -4u^4 + 8u^3  - \frac{24}{5}u^2 + \frac{8}{25}u + \frac{44}{125} &  \mbox{if} &  \frac{1}{5} \leq u < \frac{3}{5}\\
   u^4 - 4u^3 + 6u^2 - 4u + 1 &  \mbox{if} &  \frac{3}{5} \leq u \\
   \end{array}
   \right.
   \nonumber
-\end{equation}
+\end{align}
 
 
 \subsubsection{Quintic spline ($M_6$) kernel}
@@ -166,15 +153,16 @@ The kernel function $f(u)$ reads:
 In 3D, we have $C=\frac{2187}{40\pi}$ and $\gamma=H/h = 2.195775$.\\
 The kernel function $f(u)$ reads:
 
-\begin{equation}
-  M_6(u) = \left\lbrace\begin{array}{rcl}
+\begin{align}
+  M_6(u) &=     \nonumber\\
+  &\left\lbrace\begin{array}{rcl}
   -10u^5 + 10u^4 - \frac{20}{9}u^2 + \frac{22}{81} & \mbox{if} & u < \frac{1}{3} \\
   5u^5 - 15u^4 + \frac{50}{3}u^3 - \frac{70}{9}u^2 + \frac{25}{27}u + \frac{17}{81} &  \mbox{if} &  \frac{1}{3} \leq u < \frac{2}{3}\\
   -1u^5 + 5u^4 - 10u^3 + 10u^2 - 5u + 1. & \mbox{if} & u \geq \frac{2}{3}
   \end{array}
   \right.
-    \nonumber
-\end{equation}
+      \nonumber
+\end{align}
 
 
 \subsubsection{Wendland C2 kernel}
@@ -194,7 +182,7 @@ In 3D, we have $C=\frac{495}{32\pi}$ and $\gamma=H/h = 2.207940$.\\
 The kernel function $f(u)$ reads:
 
 \begin{align}
-  \Psi_{i,j}(u) &= \frac{35}{3}u^8 - 64u^7 + 140u^6\\
+  \Psi_{i,j}(u) &= \frac{35}{3}u^8 - 64u^7 + 140u^6     \nonumber\\
   & - \frac{448}{3}u^5 + 70u^4 - \frac{28}{3}u^2 + 1
     \nonumber
 \end{align}
@@ -206,7 +194,7 @@ In 3D, we have $C=\frac{1365}{64\pi}$ and $\gamma=H/h = 2.449490$.\\
 The kernel function $f(u)$ reads:
 
 \begin{align}
-  \Psi_{i,j}(u) &= 32u^{11} - 231u^{10} + 704u^9 - 1155u^8\\
+  \Psi_{i,j}(u) &= 32u^{11} - 231u^{10} + 704u^9 - 1155u^8     \nonumber\\
   & + 1056u^7 - 462u^6 + 66u^4 - 11u^2 + 1
     \nonumber
 \end{align}