Adaptivity with moving grids

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