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