### [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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