[ex][model][slides] Update Model Exercise Slides

[ex][model][slides] Replace discrete eq from first model slide

[ex][model][slides] move area in fluxes

[slides] fix typo

[ex][model][slides] Change discrete eq

[ex][model][slides] Proposel for new property instructions

slides/model.md

@@ -9,29 +9,27 @@ A DuMu^x^ model is an implementation of a discretized **mathematical model**, ge
 
 ## What is a DuMu^x^ model
 Mathematical model (PDE):
-$\begin{equation*}
-\small
+$$
+\begin{aligned}
 \frac{\partial S(u)}{\partial t} + \nabla \cdot \mathbf{F}(u) = q, \quad \forall (t,\mathbf{x}) \in (0,T] \times \Omega
-\end{equation*}$
+\end{aligned}
+$$
 
 <img src=img/model_domain_discretization.png width="100%">
 
-Discrete model, e.g. using finite volumes:
-$\begin{equation*}
-\small
-|B| \frac{S_h(\mathbf{u}^{n+1}_h) - S_h(\mathbf{u}^{n}_h)}{\Delta t}  + \sum_{\sigma \in \Sigma_B} F_{B,\sigma}(\mathbf{u}^{n+1}_h) = \int_{B}  q^{n+1} \, dx, \quad \forall t_{n+1}\leq T, \; \forall B
-\end{equation*}$
-
 ## What is a DuMu^x^ model
-Discrete model, e.g. using finite volumes:
-$\begin{equation*}
+Discrete model using finite volumes:
+$$
+\begin{aligned}
 \small
-|B| \frac{S_h(\mathbf{u}^{n+1}_h) - S_h(\mathbf{u}^{n}_h)}{\Delta t}  + \sum_{\sigma \in \Sigma_B} F_{B,\sigma}(\mathbf{u}^{n+1}_h) = \int_{B}  q \, dx, \quad \forall t_{n+1}\leq T, \; \forall B
-\end{equation*}$
+|B| \frac{S_B(\mathbf{u}^{n+1}_h) - S_B(\mathbf{u}^{n}_h)}{\Delta t}  + \sum_{\sigma \in \Sigma_B} F_{B,\sigma}(\mathbf{u}^{n+1}_h) = |B|  q_B(\mathbf{u}^{n+1}_h), \quad \forall t_{n+1}\leq T, \; \forall B
+\end{aligned}
+$$
 
-* $S_h:$ storage term
+* $B:$ control volume
+* $S_B:$ storage term
 * $F_{B,\sigma}:$ flux term over sub control volume face (scvf)
-* $q:$ source term
+* $q_B:$ source term
 
 Where to implement these terms in DuMu^x^?
 
@@ -51,9 +49,11 @@ Implements terms of a PDE in the functions
 ## Example: Diffusion equation
 Mathematical model (PDE):
 
-$\begin{equation}
-\frac{\partial c}{\partial t} - \nabla \cdot (D \nabla c) = 0 \:\: \mathrm{in}\; \Omega \times (0,T] \\
-\end{equation}$
+$$
+\begin{aligned}
+\frac{\partial c}{\partial t} - \nabla \cdot (D \nabla c) = 0 \:\: \mathrm{in}\; \Omega \times (0,T]
+\end{aligned}
+$$
 
 with
 
@@ -65,9 +65,11 @@ with
 ## Example: Diffusion equation
 Discrete model using the Box discretization:
 
-$\begin{equation}
-\vert B \vert \frac{c_B^{n+1}-c_B^{n}}{\Delta t} - \sum_{\sigma \in \Sigma_{B}} \left[ D \nabla c_h^{n+1} \cdot \boldsymbol{n}_{B,\sigma} \vert \sigma \vert \right] = 0,
-\end{equation}$
+$$
+\begin{aligned}
+\vert B \vert \frac{c_B^{n+1}-c_B^{n}}{\Delta t} - \sum_{\sigma \in \Sigma_{B}} \vert \sigma \vert \left[  D \nabla c_h^{n+1} \cdot \boldsymbol{n}_{B,\sigma} \right] = 0,
+\end{aligned}
+$$
 
 with
 
@@ -79,11 +81,13 @@ with
 ## Example: Diffusion equation
 Discrete model using the Box discretization:
 
-$\begin{equation}
-\vert B \vert \frac{c_B^{n+1}-c_B^{n}}{\Delta t} - \sum_{\sigma \in \Sigma_{B}} \left[ D \nabla c_h^{n+1} \cdot \boldsymbol{n}_{B,\sigma} \vert \sigma \vert \right] = 0,
-\end{equation}$
+$$
+\begin{aligned}
+\vert B \vert \frac{c_B^{n+1}-c_B^{n}}{\Delta t} - \sum_{\sigma \in \Sigma_{B}} \vert \sigma \vert \left[  D \nabla c_h^{n+1} \cdot \boldsymbol{n}_{B,\sigma} \right] = 0,
+\end{aligned}
+$$
 
-<img src=img/box_scv_scvf.png width="90%">
+<img src=img/box_scv_scvf.png width="80%">
 
 ## `LocalResidual`
 The local residual of the diffusion model:
@@ -112,9 +116,11 @@ NumEqVector computeStorage(const Problem& problem,
 
 ## Flux term
 
-$\begin{equation}
-F_{B,\sigma} = -D \nabla c_h^{n+1} \cdot \boldsymbol{n}_{B,\sigma} \vert \sigma \vert
-\end{equation}$
+$$
+\begin{aligned}
+F_{B,\sigma} = - \vert \sigma \vert D \nabla c_h^{n+1} \cdot \boldsymbol{n}_{B,\sigma}
+\end{aligned}
+$$
 
 with
 
@@ -182,9 +188,19 @@ For its implementation different model-specific properties have to be set
 
 # Model properties
 
+## Model properties
+
+All properties are defined within the namespace `Dumux::Properties`.
+
+```cpp
+namespace Dumux::Properties {
+    //define all properties
+} // end namespace Dumux::Properties
+```
 ## Model type tag
 
-The property tag is an empty struct with the respective name, e.g. `DiffusionModel`
+The property type tag is an empty struct with the respective name, e.g. `DiffusionModel`. 
+All properties related to the desired model can be attributed using the property type tag.
 
 ```cpp
 namespace Dumux::Properties::TTag {
@@ -192,18 +208,11 @@ namespace Dumux::Properties::TTag {
 struct DiffusionModel {};
 } // end namespace Dumux::Properties::TTag
 ```
-## Model properties
-We can set nodel properties for the `DiffusionModel` type tag
-
-All properties are defined within the namespace `Dumux::Properties`
-```cpp
-namespace Dumux::Properties {
-    //define all properties
-} // end namespace Dumux::Properties
-```
 
 ## Defining model properties
+
 The type of the local residual is the class `DiffusionModelLocalResidual` defined from earlier
+
 ```cpp
 template<class TypeTag>
 struct LocalResidual<TypeTag, TTag::DiffusionModel>
@@ -211,7 +220,9 @@ struct LocalResidual<TypeTag, TTag::DiffusionModel>
 ```
 
 ## Defining model properties
+
 The model traits specify information about the model:
+
 ```cpp
 template<class TypeTag>
 struct ModelTraits<TypeTag, TTag::DiffusionModel>