### [handbook][disc] Correction of indices used for the box method

parent 9541a0b2
 ... ... @@ -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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