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