Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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<strong>Adaptivity</strong> <strong>with</strong> <strong>moving</strong> <strong>grids</strong> 103<br />
0.5<br />
0.25<br />
0.4<br />
0.2<br />
u(ξ,t)<br />
0.3<br />
0.2<br />
t =0<br />
u(ξ,t)<br />
0.15<br />
0.1<br />
t =10<br />
0.1<br />
0.05<br />
0<br />
0 0.2 0.4 0.6 0.8 1<br />
ξ<br />
0<br />
0 0.2 0.4 0.6 0.8 1<br />
ξ<br />
10<br />
10<br />
8<br />
8<br />
t<br />
6<br />
t<br />
t<br />
6<br />
4<br />
4<br />
2<br />
2<br />
0<br />
0<br />
−4 −2 0 2 4<br />
−5 0 5<br />
x i x i t −1/3<br />
Figure 5.1. Convergence of the solution in the computational<br />
domain to the discrete self-similar solution. Note the<br />
invariance of the solution profile in this domain. We also show<br />
the evolution of the mesh and note that this scales as t 1/3 .<br />
temporarily and spatially adaptive methods for any such computation. A<br />
survey of methods and underlying numerical theory for blow-up-type problems<br />
is given in Budd et al. (1996), <strong>with</strong> more recent references in Budd<br />
and Williams (2006), Ceniceros and Hou (2001) and Huang, Ma and Russell<br />
(2008). Mesh refinement (h-adaptive) methods have been used in some<br />
numerical studies of blow-up, including the dynamic rescaling methods, such<br />
as those described in Berger and Kohn (1988), but <strong>moving</strong> mesh methods<br />
have proved to be more effective in both one and two dimensions. This is<br />
due, in part, to the strong scaling symmetry properties of the solution close<br />
to blow-up (where the effects of boundary and initial conditions become<br />
progressively less important), and the way that this can be exploited by<br />
using the scale-invariant methods described in Section 5.1.