Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
70 C. J. Budd, W. Huang and R. D. Russell<br />
If M is constant, then equation (3.46) <strong>with</strong>out the power law scaling admits<br />
a variables-separable solution for which L t = CL n . If n > 1 then this<br />
equation has solutions which blow up in a finite time. The rescaling prevents<br />
this possibility. The operator on the left of this system is a smoothing<br />
operator, similar to the operator used in (2.14), (3.9), which reduces the<br />
stiffness of this system when it is discretized. Observe that the term θ(t) has<br />
not been included in (3.46). Indeed this term arises naturally as a constant<br />
of integration. The PMA equation (3.46) has many properties in common<br />
<strong>with</strong> the <strong>moving</strong> mesh equation (3.9). In particular, if M is independent of<br />
time then the solution of (3.41) corresponds to a stable solution of (3.46).<br />
Indeed, we have the following results.<br />
Lemma 3.6.<br />
(a) Suppose that M t = 0 and the Monge–Ampère equation (3.41) admits<br />
a (steady) convex solution P (ξ) <strong>with</strong> associated map X(ξ) forwhich<br />
H(P ) > 0, so that P satisfies the equation<br />
Then:<br />
M(∇P )H(P )=θ.<br />
(i) the PMA equation (3.46) admits a time-dependent solution<br />
Q(ξ,t)= θ1/n t<br />
+ P (ξ) (3.47)<br />
ɛ<br />
for which ∇ ξ Q = ∇ ξ P = x(ξ);<br />
(ii) the resulting mesh is locally stable.<br />
(b) If M is slowly varying, then the solution of (3.46) remains ɛ-close to a<br />
solution of (3.41) for all time.<br />
Proof. This is given in Budd and Williams (2009) and is very similar to<br />
the corresponding proof given for the stability of (3.9)<br />
It is also possible to show (Budd and Williams 2009) that, throughout<br />
the evolution of the mesh, if H(Q) and∆Q are initially positive then they<br />
stay positive for all time. This application of the maximum principle guarantees<br />
that the map generating the mesh is locally invertible for all time,<br />
and hence no mesh tangling can occur. This is a very useful feature of<br />
such r-adaptive methods, and we will see numerical examples of this in the<br />
following calculations.<br />
3.3.4. Discretizing the parabolic Monge–Ampère equation (3.46)<br />
To discretize (3.46) in space we can impose a uniform grid of mesh size ∆ξ<br />
on the computational space and assume that Q and Q t take point values