From eb9ea429d041ded0bf14e9b08222381ec31cab86 Mon Sep 17 00:00:00 2001
From: Martin Schneider <martin.schneider@iws.uni-stuttgart.de>
Date: Tue, 11 Dec 2018 17:39:30 +0100
Subject: [PATCH] [handbook][disc] Correction of indices used for the box
method
---
doc/handbook/5_spatialdiscretizations.tex | 19 ++++++++++---------
1 file changed, 10 insertions(+), 9 deletions(-)
diff --git a/doc/handbook/5_spatialdiscretizations.tex b/doc/handbook/5_spatialdiscretizations.tex
index 82ffd2673e..c772f2f1b2 100644
--- a/doc/handbook/5_spatialdiscretizations.tex
+++ b/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}
--
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