Commit eb9ea429 authored by Martin Schneider's avatar Martin Schneider

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

parent 9541a0b2
......@@ -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}
......
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