### [handbook][disc] Add first draft of TPFA description

parent aaf245dc
 ... ... @@ -224,7 +224,43 @@ We denote by $\mathcal{M}$ the mesh that results from the division of the domain where $F_{K, \sigma} \approx \int_{\sigma} \left( - \mathbf{\Lambda} \nabla u \right) \cdot \mathbf{n} \mathrm{d} \Gamma$ is the discrete flux through a face $\sigma$ of cell $K$ and $Q_k = \int_K q \mathrm{d}x$ is the integrated source/sink term. Cell-centered finite-volume schemes differ in the way how the term $(\mathbf{\Lambda} \nabla u ) \cdot \mathbf{n}$ is approximated. Using the symmetry of the tensor $\mathbf{\Lambda}$, this term can be rewritten as $\nabla u \cdot \mathbf{\Lambda}\mathbf{n}$, which corresponds to the directional derivative of $u$ in co-normal direction $\mathbf{\Lambda}\mathbf{n}$. In the following, the main ideas of the two-point flux approximation and the multi-point flux approximation methods are briefly described. Hereby, we restrict the discussion to the two-dimensional case. \subsubsection{TPFA}\label{cc_tpfa} The linear two-point flux approximation is a simple but robust cell-centered finite-volume scheme, which is commonly used in commercial software. This scheme can be derived by using the conormal decomposition, which reads \begin{equation} \mathbf{\Lambda}_K \mathbf{n}_{K, \sigma} = t_{K,\sigma} \mathbf{d}_{K,\sigma} + \mathbf{d}^{\bot}_{K,\sigma}, \quad t_{K,\sigma} = \frac{\mathbf{n}_{K, \sigma}^T \mathbf{\Lambda}_K \mathbf{d}_{K,\sigma} }{\mathbf{d}_{K,\sigma}^T \mathbf{d}_{K,\sigma}}, \; \mathbf{d}^{\bot}_{K,\sigma} = \mathbf{\Lambda}_K \mathbf{n}_{K, \sigma} - t_{K,\sigma} \mathbf{d}_{K,\sigma}, \end{equation} with the distance vector $\mathbf{d}_{K,\sigma} := \mathbf{x}_\sigma - \mathbf{x}_K$ and $\mathbf{d}_{K,\sigma}^T \mathbf{d}^{\bot}_{K,\sigma} = 0$. The same can be done for the conormal $\mathbf{\Lambda}_L \mathbf{n}_{L, \sigma}$. The $t_{K,\sigma}$ and $t_{L,\sigma}$ are the transmissibilities associated with the face $\sigma$. These transmissibilities are calculated in \Dumux by using the function \texttt{computeTpfaTransmissibility}. With these notations, it follows that for each cell $K$ and face $\sigma$ \begin{equation} \nabla u \cdot \mathbf{\Lambda}_K \mathbf{n}_{K, \sigma} = t_{K,\sigma} \nabla u \cdot \mathbf{d}_{K,\sigma} + \nabla u \cdot \mathbf{d}^{\bot}_{K,\sigma}. \end{equation} For the TPFA scheme, the second part in the above equation is neglected. By using the fact that $\nabla u \cdot \mathbf{d}_{K,\sigma} \approx u_\sigma - u_K$, the discrete fluxes for face $\sigma$ are given by \begin{equation} F_{K,\sigma} = -\meas{\sigma} t_{K,\sigma} (u_\sigma - u_K), \qquad F_{L,\sigma} = -\meas{\sigma} t_{L,\sigma} (u_\sigma - u_L). \label{eq:TPFAOneSided} \end{equation} Enforcing local flux conservation, i.e. $F_{K,\sigma}+F_{L,\sigma}=0$, results in \begin{equation} u_\sigma = \frac{t_{K,\sigma} u_K + t_{L,\sigma} u_L}{t_{K,\sigma} + t_{L,\sigma}}. \end{equation} With this, the fluxes \eqref{eq:TPFAOneSided} are rewritten as \begin{equation} F_{K,\sigma} = \meas{\sigma} \frac{t_{K,\sigma} t_{L,\sigma}}{t_{K,\sigma} + t_{L,\sigma}} (u_K - u_L), \quad F_{L,\sigma} = \meas{\sigma} \frac{t_{K,\sigma} t_{L,\sigma}}{t_{K,\sigma} + t_{L,\sigma}} (u_L - u_K). \label{eq:TPFAFlux} \end{equation} By neglecting the orthogonal term, the consistency of the scheme is lost for general grids, where $\nabla u \cdot \mathbf{d}^{\bot}_{K,\sigma} \not = 0$. The consistency is achieved only for so-called K-orthogonal grids for which $\mathbf{d}^{\bot}_{K,\sigma} = 0$. For such grids we deduce that \begin{equation} \frac{t_{K,\sigma} t_{L,\sigma}}{t_{K,\sigma} + t_{L,\sigma}} = \frac{\tau_{K,\sigma} \tau_{L,\sigma}}{\tau_{K,\sigma} d_{L,\sigma} + \tau_{L,\sigma} d_{K,\sigma}}, \label{eq:TPFAcoeffNew} \end{equation} with $\tau_{K,\sigma} := \mathbf{n}_{K, \sigma} \mathbf{\Lambda}_K\mathbf{n}_{K, \sigma}, \tau_{L,\sigma} := \mathbf{n}_{L, \sigma} \mathbf{\Lambda}_L\mathbf{n}_{L, \sigma}$, $d_{K,\sigma}:= \mathbf{n}_{K, \sigma} \cdot \mathbf{d}_{K, \sigma}$, and $d_{L,\sigma}:= \mathbf{n}_{L, \sigma} \cdot \mathbf{d}_{L, \sigma}$. This reduces, for the case of scalar permeability, to a distance weighted harmonic averaging of permeabilities. ... ...
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