Adaptivity with moving grids

110 C. J. Budd, W. Huang and R. D. Russell
x 10 5
5
4.5
4
3.5
t =0.034301361416092
|u(0,t)|
3
2.5
2
1.5
1
0.5
t =0.034301361230052
0
0 0.5 1 1.5 2 2.5
r
3 3.5 4 4.5 5
x 10 −5
Figure 5.5. The solution when |u(0,t)| = 100000 and
500000. Note the narrow width of the peak.
self-similar solution for which the mesh points X i should take the form
X i = √ T − tY i , so that
|u(0,t)|X i = Q(0)Y i
is constant in time. In Figure 5.5 we plot two solutions starting from initial
data u(r, 0) = 6 √ 2 exp(−r 2 ), which have a computed blow-up time of T =
0.0343013614215. Observe the excellent resolution of the peak even when
the peak amplitude is around 10 5 and the peak width is around 10 −5 . In
Figure 5.6 we show the computed values of W i ≡|u(0,t)|X i as functions of
τ =log(T − t) for a range in which |u| varies from 100 to 500000. These
are clearly tending towards constants, indicating that both the solution and
the mesh are evolving in a self-similar manner.
It is interesting to note that the scale-invariant methods are easy to use
and give rather better results in this case than symplectic methods described
in McLachlan (1994), despite the Hamiltonian structure of the problem.
This is because the adaptive methods give much better resolution of the
peak.
In contrast, further computations of focusing solutions of the nonlinear
Schrödinger equation are given by Ceniceros (2002). In this case the following
highly dispersive problem was studied:
iɛu t + 1 2 ɛ2 u xx + u|u| 2 =0, u(x, 0) ≡ u 0 = A 0 (x)e iS 0(x)/ɛ , (5.29)