I use the line method to solve the PDE

```
D[a[t, r]t, t]+ 3/2 1 / t * D[a[t, r]t]-
10 / t * D[a[t, r]r, r]-
10 / t * 2 / rD[a[t, r]r]+ (t / 10) ^ 4 * Sin[a[t, r]]== 0,
```

with the initial conditions

```
on[10, r] ==
4 (ArcTan[Exp[(r - R0)]]+ ArcTan[Exp[(-r - R0)]])
derivative[1, 0][a][10, r] == 0,
derivative[0, 1][a][t, ri] == 0,
```

The domain is

```
{r, ri, 3R0 + ri}, {t, 10,32}
```

from where *R0 = 10*, The theoretical value of *ri* is *0*but to avoid the singularity *1 / ri* In the PDE I choose ri as a very small number, eg. *ri = 10 ^ (- 4)*,

The solution is like "a wall" from which you move *r = R0* to *r = ri = 10 ^ (- 4)* With *t* increase, and the wall becomes steeper and wavy.

The method I use to solve the PDE is the line method

```
Method -> {"MethodOfLines", "TemporalVariable" -> t,
"SpatialDiscretization" -> {"TensorProductGrid", "Coordinates" ->
{Mygrid}}}
```

I define "mygrid" so that I can have more points in the discretization of the spatial dimension (*r*). I try to spread more points in the steep part. For example:

```
Rmiddle6m = 0.1; Rmiddle5m = 0.2; Rmiddle4m = 0.5; Rmiddle3m = 1;
Rmiddle2m = 2; Rmiddle1m = 3; Rmiddle0 = 4; Rmiddle1 = 5; Rmiddle2 = 6;
Rmiddle3 = 8;
mygrid = streamline[ Join[ri + Range[0,
Rmiddle6m*1500]/ 1500, ri + Rmiddle6m + range[1, (Rmiddle5m -
Rmiddle6m)*1500]/ 1500, ri + Rmiddle5m + range[1, (Rmiddle4m -
Rmiddle5m)*1200]/ 1200, ri + Rmiddle4m + range[1, (Rmiddle3m -
Rmiddle4m)*1000]/ 1000, ri + Rmiddle3m + range[1, (Rmiddle2m -
Rmiddle3m)*600]/ 600, ri + Rmiddle2m + range[1, (Rmiddle1m -
Rmiddle2m)*300]/ 300, ri + Rmiddle1m + range[1, (Rmiddle0 -
Rmiddle1m)*80]/ 80, ri + Rmiddle0 + range[1, (Rmiddle1 -
Rmiddle0)*50]/ 50, ri + Rmiddle1 + Range[1, (Rmiddle2 -
Rmiddle1)*30]/ 30, ri + Rmiddle2 + Range[1, (Rmiddle3 -
Rmiddle2)*20]/ 20, ri + Rmiddle3 + Range[1, (3 R0 - Rmiddle3)*10]/ 10]]
```

It works as long as the solution is not so steep. But with *t* increased (the wall becomes steeper), a warning message appears:

*The scaled local spatial error estimate of … at t = … in the direction of the independent variable r is much larger than the given error tolerance.*

I think the warning message means that I did not solve the PDE exactly. Any ideas to avoid the problem? Or is there another method than *methodoflines* to solve this PDE? Many thanks!