Adaptivity with moving grids

Adaptivity with moving grids 101
Now consider a simple centred finite difference discretization of the Lagrangian
form of the underlying PDE, which takes the form
U i+1 −U i
X i+1 −X i
− U i−U i−1
X i −X i−1
U i+1 − U i−1
˙U i =
1
2 (X + f(U i )+Ẋi
i+1 − X i−1 )
(X i+1 − X i−1 ) .
We can substitute (5.20) directly into this expression to give
αV i =
V i+1 −V i
Y i+1 −Y i
− V i−V i−1
Y i −Y i−1
1
2 (Y i+1 − Y i−1 )
+ f(V i )+ Y i
2
V i+1 − V i−1
(Y i+1 − Y i−1 ) . (5.22)
It is immediately obvious that (5.22) is a consistent discretization of the
ordinary differential equation (5.21) so that the function v(y) will be approximated
by V i at the (time-independent computational) mesh point Y i .
Note further that the discretization error in approximating v by V i is independent
of the solution scale. In the case of systems such as the blow-up
problems of Section 5.2, this implies that the asymptotic form of the singularity
at the peak will be approximated with uniform accuracy for peaks
with very small spatial scales (and correspondingly large solution scales).
We note, however, that other errors do arise at points where the (rapidly)
moving mesh following the evolving peak matches a nearly stationary mesh
in the regions closer to the boundary of the domain.
To determine the location of the points Y i in terms of the computational
variables, we must apply the MMPDE used to evolve the mesh. Again, to
give an example we consider MMPDE5. Substituting (5.20) into a standard
discretization of MMPDE5 gives the following discrete equation for Y i :
1
2 Y i = 1 (
Mi+1/2
2∆ξ 2 (Y i+1 − Y i ) − M i−1/2 (Y i − Y i−1 ) ) . (5.23)
Crucially, we note that the functional equation (5.14) satisfied by the monitor
function M allows this rescaling to be made. Similar discrete equations
arise for other choices of MMPDE, provided that M satisfies (5.14).
We note that exactly the same rescalings are possible in higher-dimensional
problems using moving meshes generated by any other methods described
in this subsection, and for any other form of discretization (provided
that the processes of discretization and rescaling commute).
It is also possible to construct approximate discrete self-similar solutions
in cases where the underlying solution is better described by approximate
self-similar variables. Details of the calculations in this case are given in
Budd and Williams (2006).
As an example of this we return to the porous medium equation in one
dimension. For this problem, for all constants C>0 there is a self-similar
solution u s (x, t) of the form
u s (x, t) =(t + C) −1/3 v(y), x =(t + C) 1/3 y,