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