Adaptivity with moving grids

Adaptivity with moving grids 93
have several important properties. If carefully designed they can admit discrete
self-similar solution, they can have (local truncation) errors which are
invariant under changes in the scale of the problem, they can have discrete
conservation laws, and they can cope with symmetric singular structures.
Such problems have been studied by a number of authors: Budd et al.
(1996), Baines et al. (2006), Budd and Piggott (2005), Ceniceros and Hou
(2001), Dorodnitsyn (1991, 1993b) and Kozlov (2000).
The underlying problem that we will consider is the partial differential
equation
u t = f(u, ∇u, ∆u,...). (5.1)
Observe that this equation is invariant under translations in space and time
and also rotations in space. We also assume it to be invariant under the
action of the scaling symmetry (or symmetries)
t → λ, x → λ α x, u → λ β u. (5.2)
The objective is now to derive an adaptive method which reflects the underlying
symmetries. This can be achieved if, away from any finite boundaries,
the equations describing the mesh (regardless of whether these are
position- or velocity-based) are invariant under the action of the same symmetry
operations as the underlying PDE, so that translations in space and
time rotations and scalings of the form (5.1) leave the mesh equations invariant.
This is in many ways a very natural question to ask of an r-adaptive
method, for which the mesh may be regarded as a dynamic object, amenable
to the action of symmetries involving space and time. It is much harder to
see how h- andp-type methods can be considered in this manner.
As an example of such a problem (see also Baines et al. (2006) for the use
of an adaptive ALE method in higher dimensions, and Dorodnitsyn (1993a)
and Dorodnitsyn and Kozlov (1997) for a more abstract treatment), we
consider the one-dimensional porous medium equation given by
u t =(u m ) xx , (5.3)
where we consider this equation posed on the whole real line, with mild
decay conditions on the solution u at ∞. This partial differential equation
is invariant under two different, and independent, changes of scale, one of
space and the other of the solution. These are given by
t → λt,
x → λ 1/2 x
and
t → µt, u → µ 1/(m−1) u.
Any (linear) combination of these two changes of scale will also leave the
equation (5.3) invariant.