Adaptivity with moving grids

100 C. J. Budd, W. Huang and R. D. Russell
is invariant under the scale change Q → LQ. This implies that, if the
mesh is generated by a scale-invariant PMA-type method, then any mesh
regularity is preserved under scaling.
5.1.2. Discrete self-similar and approximately self-similar solutions
As mentioned above, a significant benefit of using a scale-invariant adaptive
scheme is that it admits discrete self-similar solutions. One reason for this
is the almost trivial, yet very important, observation that:
The actions of discretization and of rescaling commute,
where here a discretization can be any of a finite difference, collocation,
finite element or a finite volume method. (This means that a discretization
of a rescaled solution will be identical to a rescaling of a discrete solution.)
More formally, if the PDE is invariant under the action of the scaling group
t → λt, x → λ β x,u→ λ α u,
then a continuous self-similar solution takes the form
u(x,t)=t α v(y) y = x/t β . (5.19)
In terms of the computational variables, this becomes
u(x(ξ,t)) = t α v(ξ),
for appropriate functions v and y.
self-similar solution of the form
x(ξ,t)=t β y(ξ),
This leads immediately to a discrete
U i (t) =t α V i , X i (t) =t β Y i , (5.20)
For convenience we now study this discrete self-similar solution in the context
of a prototypical example, a one-dimensional system governed by a
semilinear second-order PDE of the form
u t = u xx + f(u), or in Lagrangian form u t = u xx + f(u)+ẋu x .
This PDE is invariant under the action of the scaling group
t → λt, x → λ 1/2 x, u → λ α u
(so that β =1/2), provided that the function f(u) satisfies the functional
equation
f(λ α u)=λ α−1 f(u).
Substituting the expression for the self-similar solution (5.19), and setting
β =1/2, we see that the function v(y) must satisfy the ordinary differential
equation
αv − y 2 v y = v yy + f(v), so that αv = v yy + f(v)+ y 2 v y. (5.21)