Adaptivity with moving grids

Adaptivity with moving grids 35
We now assume that the mesh function x(ξ) has regularity C 2 and exactly
equidistributes a scalar monitor function M.
Lemma 2.3.
(i) The local truncation error T of the above discretization is given in the
original variables by
[
T = ∆2 i xξξ
u xxx + u ]
xxxx
+ O(∆ 3 i ), (2.49)
3
4
x 2 ξ
and in the computational variables by
T = ∆ξ2 x 2 ξ
3
[
− M ξu xxx
+ u xxxx
Mx ξ 4
]
+ O(∆ξ 3 ). (2.50)
(ii) The truncation error is of second order if the mesh is quasi-uniform so
that condition (2.10) is satisfied.
(iii) The truncation error is zero to leading order if the mesh equidistributes
the monitor function
Proof.
M opt =(u xxx ) 1/4 =(−f x ) 1/4 . (2.51)
Let ∆ j = X j+1 − X j . A simple Taylor expansion gives
U j+1 = U j +∆ j u ′ + ∆2 j
2 u′′ + ∆3 j
6 u′′′ + ∆4 j
24 u′′′′ + O(5),
U j−1 = U j − ∆ j−1 u ′ + ∆2 j−1
2 u′′ − ∆3 j−1
6 u′′′ + ∆4 j
24 u′′′′ + O(5),
where all derivatives of u are expressed in terms of x. Hence, expanding
the left-hand side of (2.48), we obtain (after some manipulation) that the
truncation error is given by
T = 1 ( )
∆j − ∆ j−1 u ′′′ + 1 ∆ 3 j +∆3 j−1
u ′′′′ + O(3).
3
12 ∆ j +∆ j−1
This error has two components. The second is the usual component (of order
∆ 2 j ) which is seen on a uniform mesh. The first is an additional error due
to the variation in the size of the mesh. In many texts this is considered to
be large (as it is apparently of higher order); however, if ∆ j varies smoothly
over the domain then it is actually of the same order as the second error.
Since, to leading order,
∆ j =∆ξx ξ ,
we have, to leading order,
T =(∆ξ) 2 ( 1
3 x ξξu ′′′ + 1 12 x2 ξ u′′′′ )
+ O(∆ξ 3 ).