Adaptivity with moving grids

46 C. J. Budd, W. Huang and R. D. Russell
Starting from any mesh (uniform or otherwise), we can evolve the mesh by
solving MMPDE0 (3.5) or MMPDE1. Unfortunately, a uniform mesh does
not necessarily satisfy the equidistribution equation (3.1). Furthermore,
even if a mesh does exactly satisfy it at some time, solving a discretized form
of (3.5) inevitably leads to meshes that drift away from an equidistributed
state. Both of these can lead to problems with mesh crossing (the onedimensional
version of mesh tangling) which occurs when x ξ =0.
As an example of this, which also demonstrates the general applicability
of the method, we consider solving (3.5) starting from an initially uniform
mesh on [0, 1] for which x ξ = 1. In this calculation we will assume that
we have a time-evolving monitor function M(x, t) withM(x, 0) ≡ M 0 (x).
It follows from integrating (3.5) with respect to t and applying the initial
conditions that, for all time, we have
(Mx ξ ) ξ = Mξ 0 .
Hence, integrating again we have
Mx ξ = M 0 + B(t),
for some function B(t). This can be determined by integrating this expression
with respect to ξ to give
Mx ξ = M 0 + θ(t) − θ(0).
As M 0 > 0 it follows that if θ is increasing in time then x ξ > 0. However, if
θ decreases with t then it is quite possible for x t to vanish (initially at the
point where M 0 takes its minimum value) and for mesh crossing (tangling)
to result.
Such problems can be avoided (both in one and in higher dimensions)
by introducing a relaxation time into the solution of (3.3). The philosophy
behind doing this is that the equation (3.3) need not be solved exactly to
obtain a mesh which is perfectly reasonable for any computation. What is
more important is that the mesh evolves at least as fast as any significant
features of the solution. Exactly the same philosophy applies to moving
meshes in any number of dimensions. Thus it is possible to consider meshes
which relax towards an equidistributed mesh, provided the relaxation time
is smaller than the natural time scale of the solution. Ideally the relaxation
time should be of a similar order to that of the solution evolution. This
prevents the mesh equations becoming unnecessarily stiff. Various different
forms of mesh relaxation are possible.
The most obvious way of relaxing towards an equidistributed state was
proposed by Anderson and Rai (1983), who computed the mesh through a
relaxation equation, based on considering pseudo-forces between the mesh
points, given by
ɛẋ =(Mx ξ ) ξ , (3.7)