Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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34 C. J. Budd, W. Huang and R. D. Russell<br />
through equidistributing a suitable monitor function determined during the<br />
computation.<br />
It is important to note at this stage that the error in discretizing a differential<br />
equation (or indeed in interpolating the solution to that equation or a<br />
function in general) is a combination of the error that would occur on a uniform<br />
mesh together <strong>with</strong> further errors that arise from the non-uniformity<br />
of the mesh (variation in the size of the elements) and (in more than one<br />
dimension) the mesh skewness. The latter errors have to be treated <strong>with</strong><br />
great care as they can easily dominate the truncation error on the uniform<br />
mesh and make an adapted mesh worse than useless in solving the underlying<br />
problem. However, in contrast, a common error in many numerical<br />
analysis texts is to assume either that these errors add together to give a<br />
larger error, or that, for example, the error due to the non-uniformity of<br />
the mesh is always at a lower order than the error on the uniform mesh<br />
and thus dominates the overall calculation. In fact, provided that the mesh<br />
function is suitably smooth, for example if (in one dimension) the mesh is<br />
quasi-uniform and obeys the condition (2.9), then the three errors can be<br />
at the same order when expressed in terms of 1/N p . Inan‘optimal’mesh<br />
it may be possible for the errors to cancel each other out to leading order.<br />
However, such meshes are usually very hard to construct and require<br />
a lot of aprioriinformation about the solution. Adaptive meshes generally<br />
work by bounding the leading order error (regardless of the behaviour of<br />
the underlying solution).<br />
We start by looking at both optimal and adaptive non-uniform meshes on<br />
which we can pose finite difference discretization of the Poisson equation:<br />
− d2 u<br />
= f(x). (2.46)<br />
dx2 If the mesh is a function x(ξ) of the computational variable, then in the<br />
computational domain we have<br />
− 1 ( )<br />
uξ<br />
= f(x(ξ)), J = x ξ . (2.47)<br />
J J<br />
ξ<br />
The equation (2.47) can then be discretized in the computational domain<br />
for which we use the approximations<br />
U j ≈ u(X j ), X j = x(j∆ ξ ), f j = f(X j ).<br />
A natural centred difference approximation to (2.47) then takes the form<br />
<strong>with</strong><br />
(<br />
− 2<br />
Uj+1 −U j<br />
X j+1 −X j<br />
− U j−U j−1<br />
(∆ξ) 2 = f j , (2.48)<br />
X j+1 − X j−1<br />
X j −X j−1<br />
)<br />
∆ i = X i+1 − X i .