Commit 9541a0b2 authored by Martin Schneider's avatar Martin Schneider

[handbook] Update box description

parent c0f0eec5
......@@ -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 ($x_{ij}$),
......
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