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