Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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20 C. J. Budd, W. Huang and R. D. Russell<br />
0.4<br />
0.35<br />
0.3<br />
y<br />
0.25<br />
0.2<br />
0.15<br />
0.1<br />
0.05<br />
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5<br />
Figure 2.2. An example of a mesh which has<br />
tangled whilst attempting to resolve a front.<br />
x<br />
In general, position-based methods tend to produce smoother meshes, and<br />
are much less prone to the problem of mesh tangling than velocity-based<br />
methods. Mesh tangling occurs when the lines connecting adjacent mesh<br />
points intersect. An example of this is given in Figure 2.2, in which we see<br />
an attempted calculation of a solution front for Burgers’ equation which has<br />
led to a tangled mesh through the use of an inappropriately large time step<br />
in evolving the mesh.<br />
Mesh tangling can occur either locally or globally and can often arise<br />
in Lagrangian-type methods computing solutions <strong>with</strong> high vorticity. It is<br />
associated <strong>with</strong> a local loss of inevitability of the map F or, equivalently, at a<br />
point for which Λ = det(J) = 0. If Λ is controlled throughout the evolution<br />
of the mesh, then mesh tangling can be avoided. Position-based methods<br />
usually try to do this (see the calculations using the optimal transport and<br />
MMPDE methods), hence their robustness to mesh tangling. Of course,<br />
control of Λ for all time is impossible, as any equations describing the time<br />
evolution of F will inevitably be discretized in time. If this discretization is<br />
too coarse then mesh tangling may result.<br />
Mesh racing is related to mesh tangling and occurs if v is too large relative<br />
to the evolutionary behaviour of the underlying system being solved (so that<br />
the mesh evolves more rapidly than the solution of the underlying PDE).<br />
Mesh racing can occur for a variety of reasons, such as an inappropriate