Adaptivity with moving grids

Adaptivity with moving grids 85
Ω P is constant (typically unity). If A is an arbitrary measurable set in Ω C
it follows that
∫ ∫
I = dξ = M dx,
A
where B = F (A). Now, even if A is fixed, then, as the mesh is moving the
set B will typically change with time, with the points on the boundary of B
moving with velocity v. An application of the Reynolds transport theorem
implies that
d
dt
∫
B
∫
M dx =
B
B
∫
∫
M t dx + Mv · dS = (M t + ∇ · (Mv)) dx.
∂B
B
However, as A is fixed, it follows that dI/dt = 0. Furthermore, the set A
is arbitrary. It follows that M and hence v must satisfy the (geometric)
conservation law
M t + ∇ · (Mv) =0. (4.3)
If M is known, then (4.3) gives an equation for the (mesh) velocity v.
This equation must be augmented with the boundary condition
v · n on ∂Ω P . (4.4)
The equations (4.3) and (4.4) have a unique solution in one dimension but
many solutions in higher dimensions. To determine v uniquely, additional
conditions must be imposed. In the derivation of the optimal transport
methods we saw the use of an additional condition on the curl of the solution
in the computational space. In the various forms of the geometric
conservation law (GCL) methods the curl of the velocity is imposed in the
physical space. In particular, for a suitable weight function w and a background
velocity field u, the condition
∇ × w(v − u) (4.5)
is imposed, so that for an appropriate potential function φ we have
v = u + 1 w ∇φ.
Here φ is unknown, and we presume that u is specified in advance. Substituting
into the conservation law, it then follows that φ satisfies the elliptic
partial differential equation
( ) M
∇ ·
w ∇φ = −M t − ∇ · (Mu), (4.6)
with the boundary condition on ∂Ω P given by
∂φ
= −wu · n.
∂n
This equation is a scalar linear equation for φ when posed (and solved) in