Adaptivity with moving grids

94 C. J. Budd, W. Huang and R. D. Russell
Whilst the PDE may be invariant under the action of a symmetry group,
not all of the solutions have this property, although any one solution can
be mapped into another by the action of the symmetry operator. Those
solutions which are themselves invariant under the action of the symmetry
operator are termed the self-similar solutions. A self-similar solution
satisfies the functional equation
u(λt, λ α x)=λ β u(t, x). (5.4)
Such a solution can be described in terms of a new set of coordinates,
typically of the form
u(t, x) =t β v(z), z = x/t β . (5.5)
In the case of the porous medium equation, the self-similar solution with
constant mass takes the form
u(t, x) =t −1/3 v(x/t 1/3 ).
The function v(z) generally satisfies an ordinary differential equation, which
is much simpler than the original PDE. The solutions of this ODE which
correspond to solutions of the PDE are generally those with certain decay or
boundedness conditions as |z| →∞. This allows the construction of exact
solutions to PDEs in many cases. In the example of the porous medium
equation we have the famous Barenblatt–Pattle solution (Barenblatt 1996),
given by
u(x, t) =t −1/3 (a − x 2 /t 2/3 ) + ,
where a is a constant. Significantly such solutions are globally attracting (in
the sense of L 1 -convergence. An extensive description of such problems is
given in Barenblatt (1996) and Olver (1986).
More generally, solutions of PDEs with symmetries may also take the
form
u(t, x) =U(t)v(z), where z = x/L(t). (5.6)
Here U(t) andL(t) are appropriate solution and length scales. In the case
of self-similar solutions these are pure powers of t (or a translation of t) but
they can take more general forms. For example, in the case of the blow-up
equation,
u t = u xx + u 3 ,
with u →∞as t → T ,wehave
U(t) =(T − t) −1/2 and L(t) =(T − t) 1/2 | log(T − t)|.
Such solutions are called approximately self-similar solutions (Samarskii,
Galaktionov, Kurdyumov and Mikhailov 1973). Examples of this type of