Adaptivity with moving grids

Adaptivity with moving grids 95
behaviour also arise in the nonlinear Schrödinger equation in dimension two,
and in the chemotaxis equations of mathematical biology.
The self-similar (and approximately self-similar) solutions of PDEs can
play an important role in the description of the solution (beyond the fact
that they lead to exact solutions). This is because they often describe very
well the intermediate asymptotics of the solution, which is the behaviour of
the solution after the transient effects of any initial conditions and before
boundary terms become important. They are also often effective in describing
certain singular types of behaviour such as the peaks in the blow-up
and related problems, and also the interfaces in various problems in gas
dynamics (Barenblatt 1996). It is therefore useful to have numerical methods
which can accurately reproduce self-similar behaviour when it arises
in applications. One method that has been used is to make an apriori
choice of (self-similar) variables, so that the PDE can be reduced to an
ODE and to then solve this ODE numerically. This method, however, has
a number of disadvantages. Firstly, it cannot deal with general initial and
boundary conditions satisfied by the PDE. Secondly, there may often be
several symmetry groups acting on a partial differential equation and it
may often not be at all clear which (if any) leads to a self-similar solution.
Indeed, there are many problems (for example the heat equation posed on
a finite interval with an initially highly localized solution) which may have
one form of self-similar behaviour for part of the evolution and another over
longer times (for example when the solution of the heat equation interacts
with the boundary). Problems of this form (called type II self-similar solutions
in Barenblatt (1996)) are extremely hard to analyse in advance of any
PDE calculation.
An alternative approach, which makes considerable use of the r-adaptive
methods, is to use a numerical method which admits the same scaling invariances
as the original PDE away from any boundaries. Such methods
are called scale-invariant (Budd and Piggott 2005, Baines et al. 2006) as
they perform identically under the scaling transformations.
The key to the design and implementation of such methods lies in the use
of the moving mesh partial differential equations to describe the location of
the mesh points. By an appropriate choice of monitor function it is often
possible to construct such MMPDEs to be invariant under the action of the
scaling transformations. The advantages of such methods are as follows.
• They usually have discrete self-similar solutions which inherit many of
the properties of the underlying self-similar solutions.
• If designed carefully they may work for several types of scaling symmetry
and thus can be used in the case of type II self-similarity when
the exact scaling group is not known in advance.