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