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> 7<br />
defined on a uniform mesh for the same level of accuracy. Indeed, in Section<br />
5 we will give an example of the solution of focusing behaviour in the<br />
one-dimensional nonlinear Schrödinger (Sulem and Sulem 1999) equation<br />
(which can be written as four real first-order PDEs), where a discretization<br />
of the PDE on a set of N = 81 <strong>moving</strong> mesh points is able to resolve singular<br />
structures <strong>with</strong> a length scale of 10 −5 , and outperforms discretizations<br />
on uniform meshes <strong>with</strong> 10 5 mesh points. Hence the additional 81 auxiliary<br />
equations for the mesh gives a method which outperforms one <strong>with</strong> 10 5<br />
equations, giving a very significant cost reduction. Secondly, although the<br />
equations describing the mesh are indeed stiff in general, we do not need<br />
(in general) to solve them exactly. After all, it is the underlying solution of<br />
the PDE that we are interested in, and not the mesh on which it is solved.<br />
Thus a quite rough approximation to the solution of the <strong>moving</strong> mesh equations<br />
will often deliver a mesh more than adequate for the task of resolving<br />
the structures of the underlying PDE. Indeed, we will argue in Section 3<br />
(and demonstrate by example in Section 5) that a relaxed version of the<br />
<strong>moving</strong> mesh equations can be solved using a simple explicit method, will<br />
deliver similar performance for much stiffer equations than the meshes obtained<br />
by more computationally expensive methods. Indeed, they may well<br />
be stable, more regular, and deliver a better mesh quality than solving the<br />
exact equations for the mesh. Finally, one of the main applications of hp<br />
methods is to solve otherwise regular PDEs on irregular domains, typically<br />
<strong>with</strong> re-entrant corners, that introduce significant errors due to a lack of<br />
solution regularity at the corner. The r-adaptive methods as described in<br />
this paper are not really the right tool for this job (though see the results<br />
in Touringy (1998)). However, a combination of h and r methods may well<br />
prove optimal in this case, where the h method is used to mesh around the<br />
corner and the r method to follow any evolving solution structure. Future<br />
attempts to combine these two types of adaptive refinement in a general<br />
context should prove to be most interesting.<br />
1.3. Computation on <strong>moving</strong> meshes<br />
The problem of computing solutions of PDEs using a <strong>moving</strong> mesh method<br />
separates into three related problems.<br />
(1) As described, we need some monitor function to guide the mesh evolution,<br />
which is typically constrained either to equidistribute this function,<br />
or to relax towards an equidistributed state. In practice, whatever<br />
the choice of monitor function, some spatial (and temporal) smoothing<br />
is usually employed.<br />
(2) Having determined the monitor function, we must determine a mesh<br />
which equidistributes this in some way. The equidistribution problem<br />
itself is a nonlinear algebraic problem, and a variety of techniques have