Adaptivity with moving grids

90 C. J. Budd, W. Huang and R. D. Russell
4.3. The deformation map method
Deformation map methods take a strongly differential geometric approach
to moving mesh generation. Liao and his co-workers (Liao and Anderson
1992, Semper and Liao 1995, Liao and Xue 2006) have proposed a moving
mesh method based on deformation maps. Such maps were introduced by
Moser (1965) and Dacorogna and Moser (1990) in their study of volume
elements of a compact Riemannian manifold, to prove the existence of a
C 1 diffeomorphism with specified Jacobian. In this method, the mapping
x = F (ξ,t):Ω C → Ω P is determined from the system of equations
1
x t =
M(x,t) ∇φ(x,t) in Ω P ,
∆φ = − ∂M ∂t
in Ω P , (4.8)
∂φ
∂n =0 on ∂Ω P .
It is easy to see that (4.8) corresponds to the system (4.6), the GCL method
formulation which involves the potential function φ in the case where u =0
and w = M.
Dacorogna and Moser also use the alternative formulation in Bochev, Liao
and Pena (1996), given by
x t = ν(x,t)
M(x,t)
in Ω P ,
∇ · ν = − ∂M ∂t
in Ω P , (4.9)
∇ × ν =0 on ∂Ω P .
Note that letting ν = Mv, this becomes the div-curl system (4.3) and (4.5)
with u =0andw = M.
The equation for x t in both cases is nonlinear, and for simplicity explicit
time integration schemes are used by Liao and his co-workers. For (4.8),
the solution of the potential equation for φ is straightforward. However, an
implicit integration of (4.8) requires interpolation of the potential function
φ for values at points other than grid nodes, and the flux-free boundary
condition cannot generally be preserved. For (4.9), the div-curl system is
solved using a least-squares approach.
4.4. Some other velocity-based methods
The natural idea of moving mesh points, so that the velocity of the points
reflects the underlying dynamics of the solution, has led to many other
velocity-based methods besides those described above. In the velocitybased
methods of Anderson and Rai (1983), the mesh is moved according to