Adaptivity with moving grids

Adaptivity with moving grids 57
of the computational coordinates, i.e., M = M (ξ,η). A relaxation method
is proposed by Ceniceros and Hou (2001) to solve (3.32), which leads to a
set of moving mesh PDEs of the form
x τ = ∇ ξ · (M ∇ ξ x), y τ = ∇ ξ · (M ∇ ξ y). (3.33)
This system is significantly simpler than (3.31) and can be easily discretized.
The above equation, when discretized, is rather stiff and can also benefit
from a degree of mesh smoothing. A low-pass filter smoothing is applied to
the monitor function by Ceniceros and Hou (2001). Smoothing can also be
applied directly to the mesh itself, e.g.,
(1 − γ∆ ξ )x τ = ∇ ξ · (M ∇ ξ x), (1 − γ∆ ξ )y τ = ∇ ξ · (M ∇ ξ y), (3.34)
where γ>0 is related to M (typically, if a time step of ∆t is used then
γ =∆t max(M )). We note that in one dimension this system is exactly
that given by (3.9). The MMPDE method (3.34) in discretized form has
been used with success in a number of different applications, including Tang
(2005), Zegeling (2007), Ceniceros (2002) and some of the examples in Section
5.
The harmonic map method of Dvinsky (1991) given in (2.34) uses the
functional
I(ξ) = 1 ∫
√ ∑
det(M) (∇ξ i ) T M −1 ∇ξ i dx, (3.35)
2 Ω P i
while the method of Brackbill and Saltzman (1982) (cf. (2.35)) takes the
form
∫
∫
∑
I(ξ) =θ a w|J| dx + θ s (∇ξ i ) T ∇ξ i dx (3.36)
Ω P Ω P i
∫
+ θ o ((∇ξ i ) T ∇ξ j ) 2 dx.
Ω P
∑
Following Winslow (1967), Thompson et al. (1985) use a system of elliptic
differential equations for generating body-fitted, adaptive meshes. They
propose using the Poisson equations
∇ 2 ξ i = P i (x)
to control the mesh concentration and direction, where P i ,1≤ i ≤ d, are
control functions. The system can be interpreted as the Euler–Lagrange
equation of the quadratic functional
∫
I(ξ) = (|∇ξ i | 2 − P i ξ i )dx. (3.37)
Ω P
∑
i
Knupp and his co-workers (Knupp 1995, Knupp 1996, Knupp and Robidoux
2000, Knupp et al. 2002) determine the coordinate transformation
i≠j