# src/advection.h

# An advection solver

We wish to solve the advection equations $${\partial}_{t}{f}_{i}+\mathbf{\text{u}}\cdot \nabla {f}_{i}=0$$ where $\mathbf{\text{u}}$ is the velocity field and ${f}_{i}$ are a list of passive tracers. This can be done with a flux-based advection scheme such as the 2nd-order, unsplit, upwind scheme of Bell-Collela-Glaz, 1989.

The main time loop is defined in run.h. A stable timestep needs to respect the CFL condition.

```
#include "run.h"
#include "timestep.h"
```

We allocate the (face) velocity field. For compatibility with the other solvers, we allocate it as `uf`

and define an alias. The `gradient`

function is used to set the type of slope-limiting required. The default is to not use any limiting (i.e. a purely centered slope estimation).

```
face vector uf[];
#define u uf
double (* gradient) (double, double, double) = NULL;
```

Here we set the gradient functions for each tracer (as defined in the user-provided `tracers`

list).

```
extern scalar * tracers;
event defaults (i = 0) {
for (scalar f in tracers)
f.gradient = gradient;
}
```

We apply boundary conditions after user initialisation.

```
event init (i = 0) {
boundary ((scalar *){u});
boundary (tracers);
}
```

The timestep is set using the velocity field and the CFL criterion. The integration itself is performed in the events of tracer.h.

```
event velocity (i++,last) {
dt = dtnext (timestep (u, DT));
}
#include "tracer.h"
```