Adaptivity with moving grids

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