Adaptivity with moving grids

96 C. J. Budd, W. Huang and R. D. Russell
• They can be applied to problems with arbitrary initial and boundary
conditions
• They can have relative truncation errors which are independent of the
scale of the solution (Budd, Leimkuhler and Piggott 2001, Baines et al.
2006)
We give a partial proof of these results presently. A further, but less general,
advantage of such methods is that they also often preserve the asymptotic
properties of the (approximately) self-similar solutions. This is seen both
in the global convergence towards the self-similar solutions of the porous
medium equation, and the local convergence towards the singular profile
described by the approximately self-similar solution of the blow-up equation.
We start this calculation by looking at a moving mesh method in one
dimension for which the moving mesh PDE is given by MMPDE1 and for
which the monitor function is a function of u and u x , so that
(M(u, u x )x ξ ) ξ =0.
If this is to be invariant under the action of the scaling symmetry t →
λt, x → λ β x, u → λ γ u, we require that
(M(λ γ u, λ γ−β u x )λ β x ξ ) ξ =0.
This is satisfied (for all β) provided that the monitor function satisfies the
functional equation
M(λ γ u, λ γ−β u x )=λ θ M(u, u x ), (5.7)
where θ is arbitrary. Many monitor functions do not satisfy this functional
equation, for example the simple arclength monitor M = √ 1+u 2 x (although
it approximately satisfies it when |u x | is large). However, it is certainly
possible to find functions that do, and a simple example is given by
M(u, u x )=u δ
for some choice of δ. Observe that this monitor function is invariant under
a very arbitrary set of scaling symmetries and using it poses no explicit
aprioriscaling on the solution. It is thus very useful when considering
self-similar solutions of type II.
As an example we consider the porous medium equation in the form
u t =(uu x ) x |u| →0 as |x| →∞. (5.8)
It is easy to see that the first integral of this solution is constant, and we
may scale the solution so that, for all t, wehave
∫ ∞
−∞
u dx =1.