# sandbox/Antoonvh/timeaccuracy.c

# The Navier-Stokes solver is second-order accurate in time

Let us test that for a 1D viscous-flow problem, evaluated on a 2D grid.

## set-up

The set-up consists of a periodic (left-right) channel with no-slip boundary conditions (top-bottom). The initialized flow is:

$$\stackrel{\u20d7}{{u}_{0}}(x,y)=({u}_{0},{v}_{0})=(\mathrm{\text{cos}}(y),0),$$

for $\mathrm{\text{abs}}(y)\le \pi /2$. Due to the fluid’s viscousity ($\nu $), momemtum will diffuse over time, according to the (Navier-)Stokes equation, that is solved by;

$$\stackrel{\u20d7}{u}(t)=\stackrel{\u20d7}{{u}_{0}}{e}^{-\nu t}.$$

This equation will help to determine the error due to the time integration.

```
#include "grid/multigrid.h"
#include "navier-stokes/centered.h"
const face vector muv[]={1.,1.};
double timestepp;
int j;
int main(){
```

The grid resolution is chosen so fine that the presented errors here are dominated by the time stepping and are not significantly affected by the spatial discretization.

```
init_grid(1<<9);
periodic(left);
u.t[top]=dirichlet(0);
u.t[bottom]=dirichlet(0.);
μ=muv;
L0=M_PI;
X0=Y0=-L0/2.;
```

Six experiments are run, the zeroth one ($j=0$) is not used because the timestepper thinks it has made a timestep with $\Delta t=0$ for the -1-th timestep.

```
for (j=0;j<7;j++){
timestepp=2.*pow(0.5,(double)j);
run();
}
}
event init(t=0){
CFL=10000.;
foreach()
u.x[]=cos(y);
boundary(all);
DT=timestepp;
}
```

This event is used to let the run have an equidistant timestep for $j>0$.

```
event setDT(i++){
if (i==0){
DT=1.1*timestepp/0.1;
}else{
DT=timestepp;
}
}
```

## Output

After a single ‘$1/e$’ timescale, the error in the numerically obtained solution is evaluated.

```
event check(t=1.){
static FILE * fp = fopen("resultvisc.dat","w");
double err=0.;
foreach()
err+=fabs(u.x[]-cos(y)*exp(-t))*sq(Δ);
if (j>0)
fprintf(fp,"%d\t%g\n",i,err);
}
```