Adaptivity with moving grids

Adaptivity with moving grids 111
1.2
1
xi(t) |u(0,t)|
0.8
0.6
x 13
0.4
0.2
0
5 10 15 20 25 30
|log(T − t)|
x 4
x 3
x 2
Figure 5.6. The evolution of the mesh as the blow-up
time T is approached. Observe that the product
X i |u(0,t)| converges to a constant, indicating that both
solution and mesh are evolving in a self-similar manner.
with ɛ ≪ 1. In this calculation a moving mesh method with smoothing was
used, as described in Section 3. This took the form
(1 − γ∂ξξ 2 )ẋ =(Mx ξ) ξ ), (5.30)
with γ = max(M). In Ceniceros (2002) the Lagrangian form of the PDE
(5.29) was discretized in the computational domain and solved alternatively
with the MMPDE (5.30) using the methods described in Section 3. In
particular, a semi-implicit BDF method was used for the time integration
of both the moving mesh equation and the underlying PDE. A monitor
function of the form
√
M = 1+β 2 |u ξ | 2 + α 2 |u| 4
was employed, with
β =(2L) 2 ‖u 0,x‖ ∞
‖u 0 ‖
and L a measure of the size of the computational domain. In this calculation,
particular care had to be taken with the spatial discretization of the mesh
movement terms of the form ẋu x , arising in the Lagrangian form of (5.29) as
the dispersive nature of (5.29) meant that there was no natural damping to
the high-frequency terms in the errors associated with a central difference