Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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98 C. J. Budd, W. Huang and R. D. Russell<br />
arising in shocks or localized singularities, it is usually necessary to use a<br />
more sophisticated <strong>moving</strong> mesh equation to avoid mesh instabilities.<br />
In one dimension two possible equations for the mesh are the <strong>moving</strong><br />
mesh PDEs MMPDE5 and MMPDE6, given by<br />
ɛx t =(Mx ξ ) ξ , −ɛx ξξt =(Mx ξ ) ξ . (5.12)<br />
Suppose that these are used to solve a system which has intrinsic solution,<br />
length and time scales given respectively by U, L and T and a derived scale Λ<br />
for the monitor function M. As the computational variable ξ is independent<br />
of scale, the left-hand side of each of these two MMPDEs scales as L/T and<br />
the right-hand side as ΛL. These two balance (so that the mesh evolves at<br />
the same rate as the underlying solution) provided that<br />
1<br />
=Λ. (5.13)<br />
T<br />
Observe that this condition is independent of the spatial length scale L. The<br />
implication of this is that if the monitor function M(u, x) depends upon u<br />
and x, then this satisfies (5.13) provided that<br />
M(Uu,Lx)= 1 M(u, x). (5.14)<br />
T<br />
This is a more severe condition than the condition (5.14) for MMPDE1,<br />
but is necessary to ensure that the mesh calculation does not destabilize the<br />
calculation of the solution of the PDE. It has the useful property that it often<br />
gives a precise characterization of the necessary form of the monitor function<br />
appropriate to one scaling transformation. A disadvantage of this approach<br />
is, however, that it may not always (or indeed generally) be possible to<br />
capture all possible scaling transformations in a single monitor function<br />
satisfying (5.14). Thus some aprioriknowledge of the expected solution<br />
behaviour might be necessary in this case.<br />
As an example we will consider the radially symmetric solutions of the<br />
cubic nonlinear Schrödinger equation. This equation has the form<br />
iu t + u rr + n − 1 u r + u|u| 2 =0, (5.15)<br />
r<br />
where r = |x| and n is the spatial dimension. If n ≥ 2 this can have solutions<br />
which develop singularities (in amplitude and phase) in a finite time. The<br />
equation (5.15) is invariant under the action of the scaling group<br />
t → λt, r → λ 1/2 r u → λ −1/2 u,<br />
as well as the unitary multiplicative group<br />
t → t, r → r, u → e iφ u, φ ∈ R.<br />
This system develops singularities in a finite time T <strong>with</strong> a natural time