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