Commit 0b1efc32 authored by Martin Schneider's avatar Martin Schneider

[handbook][box] Replace V_j by |B_j|

parent bb148e5b
......@@ -146,16 +146,16 @@ 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_j &j = i\\
M^{lump}_{i,j} =\begin{cases} |B_j| &j = i\\
0 &j \neq i,\\
\end{cases}
\end{equation}
where $V_j$ is the volume of the FV box $B_j$ associated with node $j$.
where $|B_j|$ 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_j \frac{\partial \hat u_j}{\partial t}
|B_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 $Q_j$ is an approximation (using some quadrature rule) of the integrated source/sink term $\int_{B_j} q \: dx$.
......@@ -166,7 +166,7 @@ flow and transport equations:
\begin{equation}
\label{eq:discfin}
V_j \frac{\hat u_j^{n+1} - \hat u_j^{n}}{\Delta t}
|B_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}
......
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