Adaptivity with moving grids

104 C. J. Budd, W. Huang and R. D. Russell
A prototype example of a blow-up system is the parabolic partial differential
equation
u t =∆u + u p ,x∈ Ω, p > 1, u| ∂Ω =0. (5.24)
This has the property of single-point blow-up in that for certain types of
sufficiently large initial data, there is a point x ∗ and a finite time T so that
u(x ∗ ,t) →∞ as t → T.
Close to the point x ∗ the solution develops a narrow peak which evolves
in an approximately self-similar manner. It is not uncommon to consider
solutions which change by ten orders of magnitude, with a reduction in the
solution length scale by a similar amount. It is essential to use an adaptive
method to capture such behaviour accurately.
Example 1.
The first example that we consider is given by
u t =∆u + u 3 , x =(x, y) ∈ Ω=(0, 1) 2 ,
u(x,t)=0, x ∈ ∂Ω, (5.25)
u(x, 0) = 5 exp(−25(x− 0.45) 2 − 25(y − 0.35) 2 ).
This problem has the natural scaling symmetry
t → λt, x → λ 1/2 x, u → λ −1/2 u,
it has a natural time scale of (T −t), and it is shown in Samarskii et al. (1973)
that the natural space and solution scales are given by the approximately
self-similar variables
L =(T − t) 1/2 | log(T − t)| 1/2 U =(T − t) −1/2 .
To compute a solution, we augment this problem with the PMA equation
(5.17) to determine the moving mesh with a monitor function of the form
M ≡ M(u). To obtain scale invariance for this system we require that M
must satisfy the function equation (5.18) so that
M((T − t) −1/2 u) 1/2 1
=
(T − t) M(u).
A simple solution of this is M(u) =u 4 . In practice this monitor function
can lead to instabilities due to placing too many points in the singular region
and not sufficiently closer to the boundary of the domain, and to overcome
this we apply a McKenzie regularization to give
∫
M(u) =u(x,t) 4 + u(x ′ ,t) 4 dx ′ .
Ω P
This choice of monitor leads to a mesh which automatically inherits the correct
dynamic length scale of the underlying solution in the singular regions