Adaptivity with moving grids

Adaptivity with moving grids 97
It is then natural to choose M = u so that the monitor function has unit
integral over the physical domain. It follows immediately that
Mx ξ =1. (5.9)
Furthermore, from the geometric conservation law given by
we have
M t +(Mẋ) x =0,
u t +(uẋ) x =0.
Substituting for u t and integrating gives
u(u x +ẋ) =0,
so that the Lagrangian equation for the mesh points is given by
ẋ = −u x . (5.10)
This equation is used in Dorodnitsyn (1993a) and Dorodnitsyn and Kozlov
(1997) as the equation of motion of all of the mesh points. Note that this is
also the equation for the movement of the leading edge of the front of those
solutions of the porous medium equation which have compact support. The
same monitor function is used in Baines et al. (2006) to compute solutions
of the porous medium equation using the scale-invariant ALE method.
In the usual manner, either of the equations (5.9) and (5.10) can be
discretized and solved simultaneously with the porous medium equation
(5.8) (see Budd, Collins, Huang and Russell (1999b) for more details), so
that the discrete solution and mesh points are given by
U i ≈ u(X i ,t),
X i = x(i∆ξ).
A similar procedure can also be used with a variational formulation in higher
dimensions (Baines et al. 2006)
It is immediate (see Budd et al. (1999b)) that any such discretization
admits a discrete self-similar solution of the form
U i = t −1/3 V i , X i = t 1/3 Z i . (5.11)
It can also be shown (Budd and Piggott 2005) that such self-similar solutions
are not only locally stable, but are also global attractors, so that they
correctly organize the qualitative long-term dynamics of the solution and
even obey a discrete maximum principle.
5.1.1. Construction of scale-invariant MMPDEs
The porous medium equation has relatively benign dynamics, which allows
us to use a relatively simple moving mesh equation to evolve the mesh. In
the case of systems with more extreme forms of dynamics, such as that