# src/test/sag.c

# The SAG equation

The SAG equation is a Fick diffusion equation of a specie $c$ $$\frac{\partial c}{\partial t}={\nabla}^{2}c$$ which can be solved with the reaction–diffusion solver.

```
#include "grid/multigrid.h"
#include "run.h"
#include "diffusion.h"
```

Concentration at time $t+dt$, $t$ and maximum difference between the two.

```
scalar c[], cold[];
double errmax;
```

The generic time loop needs a timestep. We will store the statistics on the diffusion solvers in `mgd`

.

```
double dt;
mgstats mgd;
```

A “crystal growth” boundary condition of the form $$\frac{\partial c}{\partial y}=bi\phantom{\rule{0.278em}{0ex}}c\phantom{\rule{0.278em}{0ex}}\phantom{\rule{0.278em}{0ex}}\mathrm{\text{on}}\phantom{\rule{0.278em}{0ex}}\phantom{\rule{0.278em}{0ex}}y=0$$

is imposed for $-1<x<1$ and $y=0$, where $bi$ is a kind of Biot number. This mixed condition can be discretised to second-order accuracy as

`(c[] - c[bottom])/Δ = bi*(c[] + c[bottom])/2.;`

The rest of the wall is covered with a mask so that no growth occurs i.e. $\frac{\partial c}{\partial y}=0$ (a Neumann boundary condition). Combining everything then gives

```
double bi = 1;
c[bottom] = fabs(x) < 1 ? neumann(0) : c[]*(2. - bi*Δ)/(2. + bi*Δ);
```

On the top boundary the species concentration is imposed.

`c[top] = dirichlet(1);`

And there is no flux on the right and left boundaries.

```
c[right] = neumann(0);
c[left] = neumann(0);
```

## Parameters

The domain is the square box $[-5:5]\times [0:10]$.

```
int main() {
L0 = 10.;
X0 = -L0/2;
N = 256;
errmax = 5.e-3;
run();
}
```

## Initial conditions

In the absence of a mask (i.e. with the Biot condition along the bottom boundary) an exact stationary solution is

`#define cexact(y) ((bi*y + 1)/(bi*L0 + 1))`

We use this as initial condition.

```
event init (i = 0) {
foreach() {
c[] = cexact(y);
cold[] = c[];
}
boundary ({c, cold});
}
```

## Time integration

`event integration (i++) {`

We first set the timestep according to the timing of upcoming events. We choose a maximum timestep of 0.2 which ensures the stability of the reactive terms for this example.

` dt = dtnext (0.2);`

We use the diffusion solver to advance the system from $t$ to $t+dt$.

` mgd = diffusion (c, dt);`

If the difference between *c* and *cold* is small enough, we stop.

```
double dc = change (c, cold);
if (i > 0 && dc < errmax)
return 1; // stop
printf ("%g %g\n", t, dc);
}
```

## Results

```
event profiles (i += 10) {
FILE * fpx = fopen("cutx.txt", "w");
FILE * fpy = stderr;
```

The flux along $y$ is $\partial c/\partial y$

```
scalar flux[];
foreach()
flux[] = (c[0,0] - c[0,-1])/Δ;
boundary ({flux});
for (double x = -L0/2 ; x < L0/2; x += L0/N) {
double y = x + L0/2;
fprintf (fpx, "%g %g %g\n",
x, interpolate (c, x, 0) , interpolate (flux, x, 0));
fprintf (fpy, "%g %g %g\n",
y, interpolate (c, 0, y) , interpolate (flux, 0, y));
}
fclose(fpx);
}
```

At the end of the simulation, we create an image of the error field, in PNG format.

```
event pictures (t = end) {
scalar dce[];
foreach()
dce[] = c[] - cexact(y);
boundary ({dce});
output_ppm (dce, file = "c.png", spread = 2, linear = true);
}
```

We compare the exact and computed solutions for a cross-section at $x=0$.

We also display the concentration flux divided by the reference flux $bi/(biL0+1)$.

And the error field.

## Bibliography

N. Dupuis, J. Décobert, P.-Y. Lagrée, N. Lagay, D. Carpentier, F. Alexandre (2008): “Demonstration of planar thick InP layers by selective MOVPE”. Journal of Crystal Growth issue 23, 15 November 2008, Pages 4795-4798.

N. Dupuis, J. Décobert, P.-Y. Lagrée , N. Lagay, C. Cuisin, F. Poingt, C. Kazmierski, A. Ramdane, A. Ougazzaden (2008): “Mask pattern interference in AlGaInAs MOVPE Selective Area Growth : experimental and modeling analysis”. Journal of Applied Physics 103, 113113 (2008).