Adaptivity with moving grids

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