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