Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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6 C. J. Budd, W. Huang and R. D. Russell<br />
and are now well established in many commercial codes. There is a significant<br />
body of analysis supporting their use. In contrast, r-adaptive methods<br />
are more recent and are less well understood. A significant criticism which<br />
has often been made of them is that their implementation usually requires<br />
the solution of auxiliary partial differential equations for the mesh, which<br />
must be solved in parallel to the underlying partial differential equation.<br />
This requires significant additional computational cost. Furthermore, the<br />
equations to be solved to determine a suitable mesh can often be very stiff,<br />
and thus expensive to solve. Furthermore, the methods get the best estimates<br />
for a given N rather than errors necessarily lower than a specified<br />
tolerance. However, r-adaptive methods do have significant advantages in<br />
certain applications. Firstly, from a computational point of view, it is convenient<br />
to work <strong>with</strong> the same number of mesh points and the same mesh<br />
topology. This makes the linear algebra rather easier, as the matrices considered<br />
have a constant sparsity structure, and there is no need for any form<br />
of nested data structure to keep track of the node points (an issue which<br />
always complicates the use of h-refinement methods). The discretization<br />
strategy on the mesh is also easier, especially <strong>with</strong> a finite element method,<br />
as the constancy in the mesh topology and connectivity implies that there is<br />
no possibility of hanging nodes. There are further, structural advantages to<br />
r-refinement methods. One of these is that the movement of the mesh points<br />
may well correspond to natural structures of the PDE itself. An obvious example<br />
is Lagrangian-based methods for fluid flow problems, in which mesh<br />
points move <strong>with</strong> the fluid flow. A further such example is given by the use<br />
of r-refinement methods for PDEs <strong>with</strong> natural scaling symmetries, in which<br />
the mesh points automatically follow the motion of natural similarity variables<br />
(and indeed the use of the r-refinement method becomes equivalent<br />
to the use of an appropriate coordinate transformation). A third advantage<br />
of r-refinement is that, under certain circumstances, the adaptive strategy<br />
when coupled <strong>with</strong> the PDE can be regarded as one (large) dynamical system,<br />
which may then be amenable to a combined analysis. One limitation<br />
of having a fixed number of points means that it may never be possible to<br />
resolve all of the fine structures of a PDE as it evolves (although it is surprising<br />
what can be done <strong>with</strong> often a relatively small number of mesh points).<br />
Also, all r-adaptive methods are, in principle, prone to mesh tangling, in<br />
which lines connecting the mesh points can cross over during the evolution.<br />
This generally leads to severe instabilities in the system and a failure of<br />
the solution routine. Mesh tangling is often associated <strong>with</strong> mesh racing,<br />
where some mesh points move very fast during the evolution, frequently<br />
leading to a stiff set of equations to solve. The disadvantages of having to<br />
solve an auxiliary set of partial differential equations are less severe than<br />
they might originally appear to be. Firstly, the combined system of mesh<br />
and underlying equations may be much smaller than the original system