Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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36 C. J. Budd, W. Huang and R. D. Russell<br />
Setting (to leading order) ∆ j =∆ξx ξ gives (2.49). Now, from the equidistribution<br />
equation we have<br />
x ξ = θ M .<br />
Hence M ξ x ξ + Mx ξξ = 0. Substituting for M in the above gives (2.50).<br />
Result (ii) follows immediately from the expression (2.49)<br />
The optimal form of M in (iii) arises from setting the leading-order term<br />
to zero and integrating. Note that this latter calculation can break down if<br />
u xxx vanishes at some point.<br />
An almost identical calculation to the above leads to the following result.<br />
Lemma 2.4.<br />
(i) If we consider using the standard central difference approximation to<br />
u x given by<br />
u x = U i+1 − U i−1<br />
,<br />
X i+1 − X i−1<br />
then the truncation error is given in the physical coordinates by<br />
[<br />
T = ∆2 i xξξ<br />
u xx + 1 ]<br />
2<br />
3 u xxx + O(∆ 3 i ), (2.52)<br />
x 2 ξ<br />
or in the computational coordinates by<br />
T = ∆ξ2 x 2 ξ<br />
2<br />
[<br />
− M ξu xx<br />
+ 1 Mx ξ 3 u xxx<br />
]<br />
+ O(∆ξ 3 ). (2.53)<br />
(ii) This error is of second order on a quasi-uniform mesh, and is zero to<br />
leading order on an ‘optimal mesh’ given when<br />
M = ( u xx<br />
) 1/3. (2.54)<br />
These calculations, both of the errors in approximating u x and u xx and<br />
of the possible optimal meshes, are revealing in a number of ways. Firstly,<br />
they show that in all such calculations there is a subtle interplay between<br />
the mesh variability and the mesh size. This is even more marked in the<br />
case of singular perturbation problems. By choosing M very carefully we<br />
can exploit this to give very high accuracy and an optimal mesh. In general<br />
this is not usually possible. Indeed this calculation requires an accurate<br />
knowledge of the third derivative of the function u.<br />
Secondly, we can also see from (2.50) the effect of choosing other types<br />
of monitor function as part of an adaptive calculation. The optimal mesh<br />
eliminates the truncation error to leading order. However, the truncation<br />
error is actually an estimate for the second derivative (<strong>with</strong> respect to x) of<br />
the solution error between the calculated solution U j and u(X j ). To obtain