[handbook] Update box description

doc/handbook/5_spatialdiscretizations.tex

@@ -67,30 +67,30 @@ this means:
 \begin{minipage}[b]{0.47\textwidth}
 \begin{equation}
 \label{eq:p}
-	\tilde p = \sum_i N_i \hat{p_i}
+	\tilde p = \sum_i N_i \hat{p}_i
 \end{equation}
 \begin{equation}
 \label{eq:v}
-	\tilde{\mathbf v} = \sum_i N_i \hat{\mathbf v}
+	\tilde{\mathbf v} = \sum_i N_i \hat{\mathbf v}_i
 \end{equation}
 \begin{equation}
 \label{eq:x}
-	\tilde x^\kappa  = \sum_i N_i \hat x^\kappa
+	\tilde x^\kappa  = \sum_i N_i \hat x_i^\kappa
 \end{equation}
 \end{minipage}
 \hfill
 \begin{minipage}[b]{0.47\textwidth}
 \begin{equation}
 \label{eq:dp}
-	\nabla \tilde p = \sum_i \nabla N_i \hat{p_i}
+	\nabla \tilde p = \sum_i \nabla N_i \hat{p}_i
 \end{equation}
 \begin{equation}
 \label{eq:dv}
-	\nabla \tilde{\mathbf v} = \sum_i \nabla N_i \hat{\mathbf v}
+	\nabla \tilde{\mathbf v} = \sum_i \nabla N_i \hat{\mathbf v}_i
 \end{equation}
 \begin{equation}
 \label{eq:dx}
-	\nabla \tilde x^\kappa  = \sum_i \nabla N_i \hat x^\kappa .
+	\nabla \tilde x^\kappa  = \sum_i \nabla N_i \hat x_i^\kappa .
 \end{equation}
 \end{minipage}
 
@@ -113,58 +113,54 @@ yields the following equation:
 \begin{equation}
 	\int_\Omega W_j \frac{\partial \tilde u}{\partial t} \: dx + \int_\Omega W_j
 	\cdot \left[ \nabla \cdot F(\tilde u) \right]  \: dx - \int_\Omega W_j
-	\cdot q \: dx = \int_\Omega W_j \cdot \varepsilon \: dx \: \overset {!}{=} \: 0 .
+	\cdot q \: dx = \int_\Omega W_j \cdot \varepsilon \: dx \: \overset {!}{=} \: 0.	
+\label{eq:weightedResidual}	
 \end{equation}
 
-Then, the chain rule and the \textsc{Green-Gaussian} integral theorem are applied.
+For standard Galerkin schemes, the weighting functions $W_j$ are chosen the same as the ansatz functions $N_j$. However, this does not yield a locally mass-conservative scheme. 
+Therefore, for the Box method, the weighting functions $W_j$ are chosen as 
+the piecewise constant functions over a
+control volume box $B_j$, i.e.
 
 \begin{equation}
-	\int_\Omega W_j \frac{\partial \sum_i N_i \hat u_i}{\partial t} \: dx
-	+ \int_{\partial\Omega}  \left[ W_j \cdot F(\tilde u)\right]
-	\cdot \mathbf n \: d\varGamma_\Omega + \int_\Omega  \nabla W_j \cdot F(\tilde u)
-	\: dx - \int_\Omega W_j \cdot q \: dx = 0
-\end{equation}
-
-A mass lumping technique is applied by assuming that the storage capacity is
-reduced to the nodes. This means that the integrals $M_{i,j} = \int_\Omega W_j \: N_i \: dx$
-are replaced by the mass lumping term $M^{lump}_{i,j}$ which is defined as:
-
-\begin{equation}
-	 M^{lump}_{i,j} =\begin{cases} \int_\Omega W_j \: dx = \int_\Omega N_i \: dx = V_i &i = j\\
-	0 &i \neq j\\
+	W_j(x) = \begin{cases}
+	          1 &x \in B_j \\
+		  0 &x \notin B_j.\\
 	         \end{cases}
+\label{eq:weightingFunctions}	         
 \end{equation}
-where $V_i$ is the volume of the FV box $B_i$ associated with node $i$.
-The application of this assumption in combination with
-$\int_\Omega W_j \:q \: dx = V_i \: q$ yields
+Thus, the Box method is a Petrov-Galerkin scheme, where the weighting functions do not belong to the same function space than the ansatz functions.
 
+Inserting definition \eqref{eq:weightingFunctions} into equation \eqref{eq:weightedResidual} and using the \textsc{Green-Gaussian} integral theorem results in
 \begin{equation}
-	V_i \frac{\partial \hat u_i}{\partial t}
-	+ \int_{\partial\Omega}  \left[ W_j \cdot F(\tilde u)\right]
-	\cdot \mathbf n \: d\varGamma_\Omega + \int_\Omega  \nabla W_j \cdot F(\tilde u)
-	\: dx- V_i \cdot q = 0 \, .
+	\int_{B_j} \frac{\partial \tilde u}{\partial t} \: dx + \int_{\partial B_j}  F(\tilde u) \cdot \mathbf n \: d\varGamma_{B_j} - \int_{B_j} q \: dx  \overset {!}{=} \: 0, 	
+\label{eq:BoxMassBlance}	
 \end{equation}
+which has to hold for every box $B_j$. 
 
-Defining the weighting function $W_j$ to be piecewisely constant over a
-control volume box $B_i$
-
+The first term in equation \eqref{eq:BoxMassBlance} can be written as
 \begin{equation}
-	W_j(x) = \begin{cases}
-	          1 &x \in B_i \\
-		  0 &x \notin B_i\\
+\int_{B_j} \frac{\partial \tilde u}{\partial t} \: dx = \frac{d}{dt} \int_{B_j} \sum_i \hat u_i N_i  \: dx = \sum_i \frac{\partial \hat u_i}{\partial t} \int_{B_j}  N_i  \: dx.
+\end{equation} 
+Here, a mass lumping technique is applied by assuming that the storage capacity is
+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.\\
 	         \end{cases}
 \end{equation}
-
-causes $\nabla W_j = 0$:
+where $V_i$ is the volume of the FV box $B_i$ associated with node $i$.
+The application of this assumption yields
 
 \begin{equation}
 \label{eq:disc1}
 	V_i \frac{\partial \hat u_i}{\partial t}
-	+ \int_{\partial B_i}  \left[ W_j \cdot F(\tilde u)\right]
-	\cdot \mathbf n  \;  d{\varGamma}_{B_i} - V_i \cdot q = 0 .
+	+  \int_{\partial B_j}  F(\tilde u) \cdot \mathbf n \: d\varGamma_{B_j} - Q_i = 0,
 \end{equation}
+where $Q_i$ is an approximation (using some quadrature rule) of the integrated source/sink term $\int_{B_j} q \: dx$.
 
-The consideration of the time discretization and inserting $W_j = 1$ finally
+Using an implicit Euler time discretization finally
 leads to the discretized form which will be applied to the mathematical
 flow and transport equations:
 
@@ -172,7 +168,7 @@ flow and transport equations:
 \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} - V_i \: q^{n+1} \: = 0
+	\;  d{\varGamma}_{B_i} - Q_i^{n+1} \: = 0.
 \end{equation}
 
 \subsection{Cell Centered Finite Volume Method -- A Short Introduction}\label{cc}
@@ -183,7 +179,7 @@ flow and transport equations:
 \caption{\label{pc:cc} Discretization of the cell centered finite volume method}
 \end{figure}
 
-The cell centered finite volume method uses the elements of the grid as control volumes.
+The cell-centered finite volume method uses the elements of the grid as control volumes.
 For each control volume all discrete values are determined at the element/control
 volume center (see Figure~\ref{pc:cc}).
 The mass or energy fluxes are evaluated at the integration points ($x_{ij}$),