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> 97<br />
It is then natural to choose M = u so that the monitor function has unit<br />
integral over the physical domain. It follows immediately that<br />
Mx ξ =1. (5.9)<br />
Furthermore, from the geometric conservation law given by<br />
we have<br />
M t +(Mẋ) x =0,<br />
u t +(uẋ) x =0.<br />
Substituting for u t and integrating gives<br />
u(u x +ẋ) =0,<br />
so that the Lagrangian equation for the mesh points is given by<br />
ẋ = −u x . (5.10)<br />
This equation is used in Dorodnitsyn (1993a) and Dorodnitsyn and Kozlov<br />
(1997) as the equation of motion of all of the mesh points. Note that this is<br />
also the equation for the movement of the leading edge of the front of those<br />
solutions of the porous medium equation which have compact support. The<br />
same monitor function is used in Baines et al. (2006) to compute solutions<br />
of the porous medium equation using the scale-invariant ALE method.<br />
In the usual manner, either of the equations (5.9) and (5.10) can be<br />
discretized and solved simultaneously <strong>with</strong> the porous medium equation<br />
(5.8) (see Budd, Collins, Huang and Russell (1999b) for more details), so<br />
that the discrete solution and mesh points are given by<br />
U i ≈ u(X i ,t),<br />
X i = x(i∆ξ).<br />
A similar procedure can also be used <strong>with</strong> a variational formulation in higher<br />
dimensions (Baines et al. 2006)<br />
It is immediate (see Budd et al. (1999b)) that any such discretization<br />
admits a discrete self-similar solution of the form<br />
U i = t −1/3 V i , X i = t 1/3 Z i . (5.11)<br />
It can also be shown (Budd and Piggott 2005) that such self-similar solutions<br />
are not only locally stable, but are also global attractors, so that they<br />
correctly organize the qualitative long-term dynamics of the solution and<br />
even obey a discrete maximum principle.<br />
5.1.1. Construction of scale-invariant MMPDEs<br />
The porous medium equation has relatively benign dynamics, which allows<br />
us to use a relatively simple <strong>moving</strong> mesh equation to evolve the mesh. In<br />
the case of systems <strong>with</strong> more extreme forms of dynamics, such as that