Added full stencil operation example

lectures/hardware/slides.qmd

@@ -86,21 +86,78 @@ show discretisation which points to boundary-data exchange
 
 <br>
 
-![](static/stencil_operations_1.png)
+![](static/stencil_operations_1.jpg)
 
 ## Stencil operations
 
 <br>
 
-$x_t(i,j)=f(x_{t-1}(i-1,j),x_{t-1}(i,j),x_{t-1}(i+1,j),x_{t-1}(i,j-1),x_{t-1}(i,j+1))$
+$x_t(i,j)=f(x_{t-1}(i-1,j),x_{t-1}(i,j),x_{t-1}(i+1,j))$
 
-![](static/stencil_operations_2.png)
+![](static/stencil_operations_2.jpg)
 
 ## Stencil operations
 
-$x_t(i,j)=f(x_{t-1}(i-1,j),x_{t-1}(i,j),x_{t-1}(i+1,j),x_{t-1}(i,j-1),x_{t-1}(i,j+1))$
+<br>
+
+$x_t(i,j)=f(x_{t-1}(i-1,j),x_{t-1}(i,j),x_{t-1}(i+1,j))$
+
+![](static/stencil_operations_3.jpg)
+
+## Stencil operations - domain decomposition
+
+$x_t(i,j)=f(x_{t-1}(i-1,j),x_{t-1}(i,j),x_{t-1}(i+1,j))$
+
+![](static/stencil_operations_4.jpg)
 
-![](static/stencil_operations_3.png)
+## Stencil operations - domain decomposition
+
+$x_t(i,j)=f(x_{t-1}(i-1,j),x_{t-1}(i,j),x_{t-1}(i+1,j))$
+
+:::: {.columns}
+
+::: {.column width="50%"}
+* looking at a single block
+* no problem when computing the central point
+:::
+
+::: {.column width="50%"}
+![](static/stencil_operations_5.jpg)
+:::
+
+::::
+
+## Stencil operations - domain decomposition
+
+$x_t(i,j)=f(x_{t-1}(i-1,j),x_{t-1}(i,j),x_{t-1}(i+1,j))$
+
+:::: {.columns}
+
+::: {.column width="50%"}
+* But what about the point on the right border?
+:::
+
+::: {.column width="50%"}
+![](static/stencil_operations_6.jpg)
+:::
+
+::::
+
+## Stencil operations - halo cells
+
+$x_t(i,j)=f(x_{t-1}(i-1,j),x_{t-1}(i,j),x_{t-1}(i+1,j))$
+
+:::: {.columns}
+
+::: {.column width="50%"}
+* Solution: Introduction of halo cells, which need to be communicated from the neighboring block
+:::
+
+::: {.column width="50%"}
+![](static/stencil_operations_7.jpg)
+:::
+
+::::
 
 ## Beyond shared memory
 If we want to scale beyond one node, what do we need in our example?