Adaptivity with moving grids

102 C. J. Budd, W. Huang and R. D. Russell
where the function v(y) is given by the Barenblatt–Pattle profile (Barenblatt
1996). It is also well known that any positive initial data lead to a solution
u(x, t), which converges towards to a self-similar solution in the sense that
t 1/3 u(t 1/3 y, t) → v(y).
Similarly, it is shown in Budd et al. (1999b) that if the monitor function
is chosen to give a scale-invariant scheme (for example M = u), then this
scheme admits a set of discrete self-similar solutions on a moving mesh
which take the form
U i (t) =(t + C) −1/3 V i , X i (t) =(t + C) 1/3 Y i .
Note that the product
W i ≡ U i X i = V i Y i
is invariant in time. In Figure 5.1 we show the results of a computation presented
in the computational domain, using a scale-invariant moving mesh
method with M = u and a centred finite difference discretization. In this
calculation the initial data at t = 0 were taken to be an irregular function,
and results are plotted at times t =0andt = 10. We also show (dashed)
two discrete self-similar solutions with values of C chosen so that they initially
lie above and below the solution. Note that the solution calculated is
sandwiched between these two functions. We also present the corresponding
mesh X i (t) and the scaled mesh Y i (t) =X i (t)/t 1/3 . It is clear from
these figures that the computed solution converges towards the discrete selfsimilar
solutions (in correspondence with the continuous theory) and that
the moving mesh tends towards the one for which Y i is constant in time so
that X i (t) scales asymptotically as t 1/3 . Similar figures for solutions of the
porous medium equation computed using a scale-invariant ALE method in
two dimensions are given in Baines et al. (2006).
5.2. Blow-up and related problems
5.2.1. Parabolic blow-up
As mentioned in Section 5.1, a significant success in the application of moving
mesh methods occurs in the study of parabolic partial differential equations
(and also of systems of PDEs) which have solutions that blow up,
so that the solution, or some derivative of it, becomes infinite in a finite
time T . We now consider such problems in a little more detail. Blow-up
in the solution often represents an important change in the properties of
the model that the equation represents (such as the ignition of a heated gas
mixture), and it is important that it is reproduced accurately in a numerical
computation. A survey of many different types of blow-up problem is presented
in Samarskii et al. (1973). Blow-up typically occurs on increasingly
small time and length scales, and hence it is usually essential to use both