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