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An invariant moving mesh scheme for the nonlinear diffusion
equation
G Collins & C J Budd
Bath
Abstract
Many PDE problems have Lie groups of transformations that leave them as well as certain quantities invariant.
These quantities can be regarded as the natural
scalings of the PDE problem and are used as similarity variables in the construction of similarity solutions.
Invariants can also be used as computational variables in the construction
of a moving mesh scheme based upon equidistribution.
In fact it is shown that there is an important commuting relation
between a discretised scale-reduced PDE and a rescaled discretised PDE when
the discretisation is on a moving mesh.
This relation only holds if the equation governing the movement of the mesh
for the scheme is invariant to the given group. Natural test problems for invariant moving
meshes are ones posed on an infinite domain.
This talk will show how the Cauchy problem for the nonlinear diffusion
problem is solved on an infinite domain using a moving mesh scheme.
The Nonlinear diffusion equation, 
, has a group of stretching transformations
that leave the PDE and a conservation law invariant. Two invariants of the
group are, 
and 
.
These invariants are also used to construct the well known
similarity solutions for the nonlinear diffusion equation.
To solve the PDE problem numerically using the method of lines, conditions
are placed on the first and last nodes of the scheme so that the mesh will
stretch in an appropriate way to follow the solution as it diffuses.
The equations governing the mesh
movement are invariant to the same group of transformations that the PDE is. It is
shown that the invariants of the PDE
are reproduced in the discretised scheme so that the mesh 
and the computed solution 
where 
and 
are constants 
.
A system of algebraic equations is obtained by replacing these
relations into the discretised scheme and the eigenvalues of the linearisation
of this algebraic system are
studied to give convergence results.
Last modified Fri Jun 21 19:19:11 GB-Eire 1996
(DBD)