Adaptivity with moving grids

92 C. J. Budd, W. Huang and R. D. Russell
solving nonlinear PDEs which can often be naturally modified for moving
mesh algorithms, in such a way that the discrete solutions share common
properties with the (unknown) analytical solutions; and to show that the
sharing of the common structures leads to efficient adaptive methods.
5.1. Symmetry, self-similar solutions and scale invariance
Many naturally occurring physical systems have an invariance under symmetries
including translations, rotations and changes of scale, and this is
reflected in the partial differential equations that describe them. The solutions
of these equations may then either be themselves invariant under
(combinations of) these symmetries, the self-similar solutions, or they can
be transformed into other solutions through the application of symmetry
operators. Conservation laws can be linked through Noether’s theorem
(Olver 1986, Dorodnitsyn 1993b) to many of the continuous symmetries.
An example of such a system is Burgers’ equation,
u t + u · ∇u = ν∆u,
which is invariant under rotations in space and translations in both space
and time. It admits travelling-wave solutions which are self-similar solutions
coupling spatial and temporal translation, with the wave speed giving the
coupling. The waves can be at any orientation, and the action of a rotation
is to map one wave to another. In other systems a rescaling of space, time
and of the solution leaves the partial differential equations governing the
system invariant. Such changes of scale were first observed by Kepler in
his studies of the solar system and summarized in his famous third law, in
which he observed that if a planetary orbit existed with a solution (in polar
coordinates) given by (r(t),θ(t)) then there was also a solution of the form
(
(λ 3 r(λ 2 t)), O(λ 2 t) ) for any positive value of λ. Such scaling invariance also
arises in the equations of fluid and gas dynamics, nonlinear optics and many
biological systems. An excellent summary of scale-invariant systems with
such properties is given in the book by Barenblatt (1996).
We can then ask the question of whether a numerical method can be
constructed which is also invariant under symmetries. More generally, an
adaptive method can be designed to exploit the symmetries in the problem.
In a sense, the answer to this question is obvious as we can always perform
an aprioriscaling of the problem and then use a method in the scaled
coordinates. However, this assumes more knowledge of the physical system
than we may easily have available to us. Indeed, for some systems there
are a number of different possible changes of scale, and it is not apriori
obvious which one correctly describes the system evolution.
We now show how several of the moving mesh strategies described earlier,
particularly those based on moving mesh partial differential equations, can
be very effectively applied to such problems with symmetry. Such methods