diff --git a/slides/model.md b/slides/model.md index 8a7bccbc1506a1d8033559d9b823d5fd6bf8a494..e7f8dfac7d81feee39748fe6721bce281219f8ab 100644 --- a/slides/model.md +++ b/slides/model.md @@ -11,7 +11,7 @@ A DuMu^x^ model is an implementation of a discretized **mathematical model**, ge Mathematical model (PDE): $$ \begin{aligned} -\frac{\partial S(u)}{\partial t} + \nabla \cdot \mathbf{F}(u) = q, \quad \forall (t,\mathbf{x}) \in (0,T] \times \Omega +\frac{\partial S(u)}{\partial t} + \nabla \cdot \mathbf{F}(u) = q(u), \quad \forall (t,\mathbf{x}) \in (0,T] \times \Omega \end{aligned} $$ @@ -71,12 +71,7 @@ $$ \end{aligned} $$ -with - -- $c_B^n:$ concentration at time $t_n$ and control volume $B$ -- $c^n_h:$ global discrete solution at time $t_n$, interpolated using __basis functions__ -- $\mathbf{n}:$ unit outer normal vector -- $\sigma:$ sub control volume face (scvf) +<img src=img/box_scv_scvf.png width="80%"> ## Example: Diffusion equation Discrete model using the Box discretization: @@ -87,7 +82,12 @@ $$ \end{aligned} $$ -<img src=img/box_scv_scvf.png width="80%"> +with + +- $c_B^n:$ concentration at time $t_n$ and control volume $B$ +- $c^n_h:$ global discrete solution at time $t_n$, interpolated using __basis functions__ +- $\mathbf{n}:$ unit outer normal vector +- $\sigma:$ sub control volume face (scvf) ## `LocalResidual` The local residual of the diffusion model: