Adaptivity with moving grids

108 C. J. Budd, W. Huang and R. D. Russell
35
35
30
30
25
25
τ
20
15
τ
20
15
10
10
5
5
0
0 0.2 0.4 0.6 0.8 1
X i (ξ,τ)
(a)
0
−15 −10 −5 0
log(X i (ξ,τ))
(b)
Figure 5.4. Example 2. Blow-up mesh. (a) Grid in the natural
coordinates. (b) Plotting τ = | log(T − t)| as a function of log(X i )
at equally spaced values of ξ, showing the contraction rate and
approximately self-similar behaviour close to the origin.
A straightforward calculation based on the scaling structures of u shows
that, in the peak region,
( ( )
1
1
u = O √
), u xx = O
T − t L 2√ ,
T − t
( )
( )
1
1
u xxx = O
L 3√ , u xxxx = O
T − t
L 4√ .
T − t
If ξ