Adaptivity with moving grids

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