Adaptivity with moving grids

112 C. J. Budd, W. Huang and R. D. Russell
discretization This is in contrast to the highly dissipative nature of the
discretization of (5.30), which took the form
X n+1
j
− Xj
n
∆t
= γ Xn+1 j+1 − 2Xn+1 j
+ Xj−1
n+1
∆ξ 2 − γ Xn j+1 − 2Xn j + Xn j−1
∆ξ 2
+ 1 [
M
n
∆ξ 2 j+1/2
(Xj+1 n − Xj n ) − Mj−1/2 n (Xn j − Xj−1) n ] .
To avoid some of the resulting instabilities reported in Li et al. (1998) and
discussed in Section 5.3 in the context of the solution of Burgers’ equation,
a high-order (fourth-order central difference) expression was used to
discretize the Lagrangian advective term ẋu x . The resulting semi-implicit
BDF discretization then took the form
1 n+1
[3Uj
− 4Uj n + Uj n−1 ]=
2∆t
i ɛσn+1 [
]
j
2∆ξ 2 σ n+1 n+1
j+1/2
(Uj − Uj n+1 ) − σ n+1 n+1
j−1/2
(Uj − Uj−1
n+1
i
+2[
ɛ |U j n | 2 Uj n U
+ Ẋn j−2 n − 8U j−1 n +8U j+1 n − U n ]
j+2
j
[ i
−
ɛ
n−1
|Uj
| 2 Uj
n−1
X n j−2 − 8Xn j−1 +8Xn j+1 − Xn j+2
+ Ẋn−1 j
U n−1
j−2
− 8U
n−1
j−1
n−1
+8Uj+1 − U j+2
n−1
Xj−2 n−1 − 8Xn−1 j−1 +8Xn−1 j+1 − Xn−1 j+2
with σ =1/x ξ . The resulting moving mesh method was then found to be
highly effective in computing the focusing solutions.
Remark. It bears mentioning that finding self-similar or approximately
self-similar structures for analytical solutions to PDEs can be extremely
demanding, and moving mesh methods with built-in scale invariance can
often be used to gain insight into the form of such structures. A case in point
is the paper by Budd, Galaktionov and Williams (2004), which analysed an
unexpected form of blow-up for a fourth-order PDE, entirely motivated by
the results of numerical calculations with a moving mesh method.
5.3. Some problems with moving fronts
A classical problem leading to the formation of sharp fronts is Burgers’
equation given (in two dimensions) by
u t + 1 2 (u2 ) x + 1 2 (u2 ) y = ν∆u, ν ≪ 1. (5.31)
This equation has been used as a benchmark for a number of different
moving mesh algorithms (Huang and Russell 1997a, Mackenzie and Mekwi
2007a, Zhang and Tang 2002, Tang and Xu 2007, Li et al. 1998).
]
,