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