Commit a900006d by Martin Schneider

### [handbook][disc] Use dx d\Gamma consistenly for integrals

parent 0b1efc32
 ... ... @@ -45,13 +45,13 @@ In the following, the discretization of the balance equation is going to be deri From the \textsc{Reynolds} transport theorem follows the general balance equation: \underbrace{\int_\Omega \frac{\partial}{\partial t} \: u \: dx}_{1} + \underbrace{\int_{\partial\Omega} (\mathbf{v} u + \mathbf w) \cdot \textbf n \: d\varGamma}_{2} = \underbrace{\int_\Omega q \: dx}_{3} \underbrace{\int_\Omega \frac{\partial}{\partial t} \: u \, \mathrm{d}x}_{1} + \underbrace{\int_{\partial\Omega} (\mathbf{v} u + \mathbf w) \cdot \textbf n \, \mathrm{d}\Gamma}_{2} = \underbrace{\int_\Omega q \, \mathrm{d}x}_{3} f(u) = \int_\Omega \frac{\partial u}{\partial t} \: dx + \int_{\Omega} \nabla \cdot \underbrace{\left[ \mathbf{v} u + \mathbf w(u)\right] }_{F(u)} \: dx - \int_\Omega q \: dx = 0 f(u) = \int_\Omega \frac{\partial u}{\partial t} \, \mathrm{d}x + \int_{\Omega} \nabla \cdot \underbrace{\left[ \mathbf{v} u + \mathbf w(u)\right] }_{F(u)} \, \mathrm{d}x - \int_\Omega q \, \mathrm{d}x = 0 where term 1 describes the changes of entity $u$ within a control volume over time, term 2 the advective, diffusive and dispersive fluxes over the interfaces ... ... @@ -106,14 +106,14 @@ of the residual $\varepsilon$ with a weighting function $W_j$ and claiming that this product has to vanish within the whole domain, \int_\Omega W_j \cdot \varepsilon \: \overset {!}{=} \: 0 \qquad \textrm{with} \qquad \sum_j W_j =1 \int_\Omega W_j \cdot \varepsilon \, \mathrm{d}x \overset {!}{=} \: 0 \qquad \textrm{with} \qquad \sum_j W_j =1 yields the following 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. \int_\Omega W_j \frac{\partial \tilde u}{\partial t} \, \mathrm{d}x + \int_\Omega W_j \cdot \left[ \nabla \cdot F(\tilde u) \right] \, \mathrm{d}x - \int_\Omega W_j \cdot q \, \mathrm{d}x = \int_\Omega W_j \cdot \varepsilon \, \mathrm{d}x \: \overset {!}{=} \: 0. \label{eq:weightedResidual} ... ... @@ -133,17 +133,17 @@ Thus, the Box method is a Petrov-Galerkin scheme, where the weighting functions Inserting definition \eqref{eq:weightingFunctions} into equation \eqref{eq:weightedResidual} and using the \textsc{Green-Gaussian} integral theorem results in \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, \int_{B_j} \frac{\partial \tilde u}{\partial t} \, \mathrm{d}x + \int_{\partial B_j} F(\tilde u) \cdot \mathbf n \, \mathrm{d}\Gamma_{B_j} - \int_{B_j} q \, \mathrm{d}x \overset {!}{=} \: 0, \label{eq:BoxMassBlance} which has to hold for every box $B_j$. The first term in equation \eqref{eq:BoxMassBlance} can be written as \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. \int_{B_j} \frac{\partial \tilde u}{\partial t} \, \mathrm{d}x = \frac{d}{dt} \int_{B_j} \sum_i \hat u_i N_i \, \mathrm{d}x = \sum_i \frac{\partial \hat u_i}{\partial t} \int_{B_j} N_i \, \mathrm{d}x. Here, a mass lumping technique is applied by assuming that the storage capacity is reduced to the nodes. This means that the integrals $M_{i,j} = \int_{B_j} N_i \: dx$ reduced to the nodes. This means that the integrals $M_{i,j} = \int_{B_j} N_i \, \mathrm{d}x$ are replaced by some mass lumped terms $M^{lump}_{i,j}$ which are defined as M^{lump}_{i,j} =\begin{cases} |B_j| &j = i\\ ... ... @@ -156,9 +156,9 @@ The application of this assumption yields \label{eq:disc1} |B_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, + \int_{\partial B_j} F(\tilde u) \cdot \mathbf n \, \mathrm{d}\Gamma_{B_j} - Q_j = 0, where $Q_j$ 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 $\int_{B_j} q \, \mathrm{d}x$. Using an implicit Euler time discretization finally leads to the discretized form which will be applied to the mathematical ... ... @@ -168,7 +168,7 @@ flow and transport equations: \label{eq:discfin} |B_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. \; \mathrm{d}\Gamma_{B_j} - Q_j^{n+1} \: = 0. Equation \eqref{eq:discfin} has to be fulfilled for each box $B_j$. ... ... @@ -193,7 +193,7 @@ We denote by $\mathcal{M}$ the mesh that results from the division of the domain For the derivation of the finite-volume formulation we integrate the first equation of \eqref{eq:elliptic} over a control volume $K$ and apply the Gauss divergence theorem: \int_{\partial K} \left( - \mathbf{\Lambda} \nabla u \right) \cdot \mathbf{n} \, \mathrm{d} \Gamma = \int_K q \, \mathrm{d}\Omega. \int_{\partial K} \left( - \mathbf{\Lambda} \nabla u \right) \cdot \mathbf{n} \, \mathrm{d} \Gamma = \int_K q \, \mathrm{d}x. \label{eq:ellipticIntegrated} ... ...
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