### New section for implicit spatial discretization methods shortly explaining the...

New section for implicit spatial discretization methods shortly explaining the box and cc method. Reviewed by Lena.

git-svn-id: svn://svn.iws.uni-stuttgart.de/DUMUX/dumux/trunk@10237 2fb0f335-1f38-0410-981e-8018bf24f1b0
parent e371876c
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 ... ... @@ -137,7 +137,7 @@ unknowns in the system description. The available primary variables are, e.\ g., saturations, mole/mass fractions, temperature, pressures, etc. \input{box} \input{spatialdiscretization} \section{Available models} ... ...
 \section{Box method - A short introduction}\label{box} \section{Implicit Spatial Discretization Schemes}\label{spatialdiscretization} For the spatial discretization the so called BOX-method is used which unites the advantages of the finite-volume (FV) and finite-element (FE) methods. For the implicit models there are two spatial discretization schemes (BOX and Cell Centered Finite Volume Method) available which are shortly introduced in this section. \subsection{Box method - A short introduction}\label{box} The so called BOX-method is unites the advantages of the finite-volume (FV) and finite-element (FE) methods. First, the model domain $G$ is discretized with a FE mesh consisting of nodes i and corresponding elements $E_k$. Then, a secondary FV mesh is constructed by connecting the midpoints and barycenters of the elements surrounding node i creating a box $B_i$ around node i (see Figure \ref{pc:box}a). ... ... @@ -118,3 +123,23 @@ The consideration of the time discretization and inserting $W_j = 1$ finally lea \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 \end{equation} \subsection{Cell Centered Finite Volume Method - A short introduction}\label{cc} \begin{figure} [h] \centering \includegraphics[width=0.4\linewidth,keepaspectratio]{EPS/cc_disc} \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. 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 ($x_{ij}$), which are located at the midpoints of the control volume faces. This is a two point flux approximation since the flux between the element/control volume centers $i$ and $j$ is calculated only with information from these two points. In contrast the BOX-method uses a multi-point flux approximation where all nodes of the element influence the flux between two specific nodes. \\ Neumann boundary conditions are applied at the boundary control volume faces and Dirichlet boundary conditions at the boundary control volumes. \\ The cell centered finite volume method is robust and mass conservative but should only be applied for structured grids (the control volume face normal vector ($n_{ij}$) should be parallel to the direction of the gradient between the two element/control volume centers).
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