Adaptivity with moving grids

106 C. J. Budd, W. Huang and R. D. Russell
the previous theory. (Away from the solution peak the mesh is less smooth;
however, in this region the solution gradients are much smaller than in the
peak, and the local truncation errors are thus lower.)
Example 2. In order to consider these results in the context of some of
the error analysis presented in Sections 2 and 3, it is instructive to look at a
second one-dimensional example, in which we consider the blow-up problem
u t = u xx + u 3 , u x (0) = u(1) = 0.
It is known (Samarskii et al. 1973) that this equation has solutions which
blow up at the origin, and in the peak the asymptotic blow-up profile of
this solution takes the form
1 1
u(x, t) = √ √
T − t 1+ax 2 /L , (5.26)
2
where a is a constant (depending weakly on the initial conditions) which we
may take equal to unity, and L(t) is the natural length scale given by
L(t) = √ (T − t)| log(T − t)|.
As this is now a problem in one dimension, we initially take M = u 2 so
that, in the peak,
1 1
M =
(T − t) 1+x 2 /L 2 .
The integral of M is overwhelmingly dominated by the contribution in the
peak, so that
∫ 1
L
θ = M dx =
0 (T − t) tan−1 (1/L) ≈ πL
2(T − t) .
We can then take a regularized monitor function of the form
¯M = M + θ,
with
∫ 1
0
¯M =2θ,
to ensure that we have a 50:50 mesh. In the peak, ¯M ≈ M and an equidistributed
mesh satisfies the equation Mx ξ =2θ so that
x ξ = πL(1 + x 2 /L 2 ).
It follows immediately that in the peak we have
x(ξ,t) =L(t)tan(πξ). (5.27)
Observe that, if ξ < 1/2, then x = O(L), and that this mesh matches
naturally to one for which x = O(1) as ξ → 1/2, so that we have (as
required) half of the mesh points in the peak and half outside the peak.