[handbook][disc] Correction of indices used for the box method

doc/handbook/5_spatialdiscretizations.tex

@@ -146,19 +146,19 @@ Here, a mass lumping technique is applied by assuming that the storage capacity
 reduced to the nodes. This means that the integrals $M_{i,j} = \int_{B_j}  N_i \: dx$
 are replaced by some mass lumped terms $M^{lump}_{i,j}$ which are defined as
 \begin{equation}
-	 M^{lump}_{i,j} =\begin{cases}  V_i &i = j\\
-	0 &i \neq j.\\
+	 M^{lump}_{i,j} =\begin{cases}  V_j &j = i\\
+	0 &j \neq i,\\
 	         \end{cases}
 \end{equation}
-where $V_i$ is the volume of the FV box $B_i$ associated with node $i$.
+where $V_j$ is the volume of the FV box $B_j$ associated with node $j$.
 The application of this assumption yields
 
 \begin{equation}
 \label{eq:disc1}
-	V_i \frac{\partial \hat u_i}{\partial t}
-	+  \int_{\partial B_j}  F(\tilde u) \cdot \mathbf n \: d\varGamma_{B_j} - Q_i = 0,
+	V_j \frac{\partial \hat u_j}{\partial t}
+	+  \int_{\partial B_j}  F(\tilde u) \cdot \mathbf n \: d\varGamma_{B_j} - Q_j = 0,
 \end{equation}
-where $Q_i$ is an approximation (using some quadrature rule) of the integrated source/sink term $\int_{B_j} q \: dx$.
+where $Q_j$ is an approximation (using some quadrature rule) of the integrated source/sink term $\int_{B_j} q \: dx$.
 
 Using an implicit Euler time discretization finally
 leads to the discretized form which will be applied to the mathematical
@@ -166,10 +166,11 @@ flow and transport equations:
 
 \begin{equation}
 \label{eq:discfin}
-	V_i \frac{\hat u_i^{n+1} - \hat u_i^{n}}{\Delta t}
-	+ \int_{\partial B_i}  F(\tilde u^{n+1}) \cdot \mathbf n
-	\;  d{\varGamma}_{B_i} - Q_i^{n+1} \: = 0.
+	V_j \frac{\hat u_j^{n+1} - \hat u_j^{n}}{\Delta t}
+	+ \int_{\partial B_j}  F(\tilde u^{n+1}) \cdot \mathbf n
+	\;  d{\varGamma}_{B_j} - Q_j^{n+1} \: = 0.
 \end{equation}
+Equation \eqref{eq:discfin} has to be fulfilled for each box $B_j$.
 
 \subsection{Cell Centered Finite Volume Method -- A Short Introduction}\label{cc}