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> 99<br />
scale of (T − t), a length scale of L =(T − t) 1/2 , and a solution scale of<br />
U =(T − t) −1/2 . For the <strong>moving</strong> mesh PDE based on a simple monitor<br />
function of the form M(u) to be invariant under the action of the unitary<br />
multiplicative group, we require that<br />
M(u) =M(|u|).<br />
The condition (5.14) then implies that<br />
M((T − t) −1/2 u)=<br />
Both conditions are satisfied if<br />
1<br />
(T − t) M(u).<br />
M(u) =|u| 2 . (5.16)<br />
Calculations using a regularized form of this monitor function are described<br />
in Section 5.1.2.<br />
Scale-invariant <strong>moving</strong> mesh methods for a general class of scale-invariant<br />
PDEs can also be constructed in higher dimensions, say x ∈ R n . We consider<br />
two cases, firstly the method of Ceniceros and Hou (2001), and then the<br />
optimal transport method. The first of these describes a two-dimensional<br />
<strong>moving</strong> mesh generated by the PDE system<br />
x t = ∇ ξ .(M∇ ξ x),<br />
y t = ∇ ξ .(M∇ ξ y).<br />
This system scales in an identical manner to both MMPDE5 and MMPDE6,<br />
and consequently is scale-invariant provided that the monitor function M<br />
satisfies the condition (5.14). In the case of optimal transport in an n-<br />
dimensional system, the PMA equation gives a mesh from ∇ ξ Q, where Q<br />
satisfies the PDE<br />
(I − γ∆)Q t =(M(u)H(Q)) 1/n . (5.17)<br />
Now, if the underlying problem has the natural scaling symmetry x → λx,<br />
then this is equivalent to the scaling symmetry Q → LQ. It is immediate<br />
that<br />
H(LQ) =L n H(Q).<br />
It then follows immediately that (5.17) is invariant under the action of the<br />
scaling symmetries provided that the monitor function satisfies the functional<br />
equation<br />
M(Uu,Lx) 1/n = 1 M(u, x). (5.18)<br />
T<br />
We note that, in two dimensions, the scaling structure of the function Q<br />
also implies that the mesh skewness s, as defined in Section 3 by the relation<br />
s = ∆(Q)2<br />
H(Q) − 2,