Adaptivity with moving grids

Adaptivity with moving grids 105
(a)
(b)
Figure 5.2. Example 1. Final grid for the blow-up example.
(a) Entire grid. (b) Detail near the blow-up point. Note that
the grid is quite regular in the vicinity of the singularity, with
no evidence of any skewness or tangling.
where u is large. We can then find a solution of the blow-up problem in the
computational domain Ω C , using a finite difference method with a uniform
N =30×30 mesh in the computational domain to discretize both the PMA
equation and the Lagrangian form of the underlying PDE. The resulting
system of ODEs was then solved simultaneously using a BDF method. In
this calculation, as the blow-up time was approached an adaptive time step
was used by applying the Sundman transformation, as described in Budd
et al. (2001). For this we take
1
∆t =
(max u) 2 .
In Figure 5.2 we show the final grid both over the whole domain and close
to the peak near the centre, as well as the initial and final solutions. The
integration was performed until |u| ∞ =10 15 , for which the peak has an
approximate length scale of 10 −15 .
Over the course of the evolution we see a mesh compression (and a solution
amplification) by a factor of 10 12 in the physical domain. Note that this
has been achieved with a very modest number of mesh points. The final
mesh shows a similar gradation of mesh elements from size O(10 −2 ) to size
O(10 −14 ). However, if we study the mesh close to the solution peak, as
illustrated in Figure 5.2, we see that this shows a strong degree of local
regularity, with no evidence of long thin elements or skewness in the region
where the solution gradients are very large. This is exactly as predicted by