Adaptivity with moving grids

Adaptivity with moving grids 47
where ɛ>0 is presumed to be small. Alternatively, we can consider the
original equidistribution equation in integral form. If a mesh is not exactly
equidistributed then we can determine the residual
R =
∫ x
a
M dx − ξ
∫ b
a
M dx.
If we then set ɛẋ = −R and differentiate this expression twice with respect
to ξ, weobtain
ɛ(ẋ) ξξ = −(Mx ξ ) ξ . (3.8)
This equation was originally derived in Adjerid and Flaherty (1986). The
equations (3.7) and (3.8) are known respectively (Huang et al. 1994) as
MMPDE5 and MMPDE6. We can combine them to give the (smoothed)
moving mesh equation considered in Huang and Russell (1997a) (seealso
the discussion in Section 2), which takes the form
ɛ
(1 − γ ∂2
∂ξ 2 )
ẋ =(Mx ξ ) ξ . (3.9)
Here γ>0 can be chosen to give some control over the smoothness of the
mesh. The equation (3.9) (and its various discretizations) is very dissipative,
and leads to extremely stable meshes under most discretizations. The
equation (3.9) also has natural extensions to higher dimensions, both in
the context of the methods described in Huang and Russell (1997a) and
Ceniceros and Hou (2001) and also in the optimal transport methods we
consider later in this section. Other smoothed versions of the MMPDEs
have also been considered by Huang and Russell (1997a). One of them is
given by
)
ɛ
(1 − γ ∂2 ∂
∂ξ 2 ∂ξ
(
−
( ∂x
∂ξ
) −2 )
)
∂ẋ
= −
(1 − γ ∂2 ∂
∂ξ
∂ξ 2 ∂ξ
( ) ∂x
. (3.10)
∂ξ
Huang and Russell (1997a) have proved that the solutions (i.e., the coordinate
transformation and the mesh) to the continuous equation (3.10) using
a central finite difference discretization have the properties both of local
quasi-uniformity and no node-crossing.
Note. Many other MMPDEs have been derived, such as MMPDE2, given by
(Mẋ) ξξ = −(M t x ξ ) ξ − 1 ɛ (Mx ξ) ξ .
However, we will focus our discussion on the more widely used (3.9).
The moving mesh equation evolves a mesh towards an equidistributed
state satisfying (3.3). When implementing the MMPDE method, the equation
(3.9) is typically discretized over the computational space, leading to a
set of ordinary differential equations for the location of the mesh points X i .