Adaptivity with moving grids

Adaptivity with moving grids 49
change too rapidly with time. As this procedure starts from a mesh
which exactly equidistributes M(X, 0), the right-hand side of (3.9) is
always close to zero and the resulting differential equations are not
especially stiff.
Observe that this algorithm has the convenience of starting from a uniform
mesh. This is a significant advantage over methods based on (3.5), or related
methods such as the GCL method described in Section 4.
We will discuss later in this section the exact mechanism by which this
algorithm for moving the mesh is coupled to the solution method for the
underlying PDE.
This procedure has been criticized (for example, see Tang (2005)) for
being imprecise about the way that ɛ is defined and the possibility of having
to solve a very stiff system of equations. However, it can be given a very
precise meaning. In the second stage of this calculation we are trying to
find a mesh which is close to an equidistributed mesh. The natural time
scale τ over which this mesh evolves is given simply by
τ ≈ ɛ M . (3.13)
The key factor governing the choices of both ɛ and M is then to ensure that
τ is smaller than but of the same order as the natural evolutionary time
scale of the underlying PDE. In Section 5 we will show that in the context
of PDEs with a strong scaling structure, this allows a natural choice to be
made for both ɛ and M.
We now substantiate some of the claims made above, as well as stating
another important property of the solutions of the moving mesh PDE (3.9).
Theorem 3.2.
(i) If M t = 0, then the equidistributed mesh is a solution of (3.9) and is
linearly stable.
(ii) If M t = O(1), then an initially ɛ-close to equidistributed solution of
(3.9) remains ɛ-close for all subsequent times.
(iii) At all times the solution of (3.9) satisfies x ξ > 0, so that mesh crossing
(tangling) does not occur.
Note. This applies for exact solutions of (3.9). If an overly coarse discretization
is used to approximate these solutions then (iii) above may be
violated (Smith 1996).
Proof. (i) Let ˆx be an equidistributed mesh satisfying (M(ˆx)ˆx ξ ) ξ =0,
with ˆx t = M t =0. It follows immediately that ɛ( ˙ˆx − γ ˙ˆx ξξ )=(M ˆx ξ ) ξ so
that ˆx satisfies the equidistribution equation. Now set x =ˆx + R(ξ,t) with