Adaptivity with moving grids

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72 C. J. Budd, W. Huang and R. D. Russell (a) t =0 (b) t =1/4 (c) t =1/2 (d) t =10 Figure 3.7. Example 1. Snapshots at t = 0 (a), t =1/4 (b), t =1/2 (c), t = 10 (d). As the solution is advected, the grid follows the maxima of the solution and does not ever fall behind or become distorted.
Adaptivity with moving grids 73 Example 2. In an example taken directly from Cao et al. (2002), we consider a monitor function localized on a sine wave of the form M 2 (x, y, t) = 1 + 5 exp(−50|(y − 1/2 − 1/4sin(2πx)sin(2πt)|), and proceed to use exactly the same method as described in Example 1. See Figure 3.8. In this example we again see that the PMA method has generated a very regular and periodic (in time) mesh. Example 3. In this example we consider a monitor function localized on the support of two travelling linear fronts moving toward each other and show the resulting mesh in Figure 3.9. For this we take x 0 = t, y 0 =0.2+t/2, u 0 = γ sech(λ(x − x 0 + y − y 0 )), x 1 =1− t, y 1 =0.8 − t/2, u 1 = γ sech(λ(x − x 1 + y − y 1 )), M 3 (x, y, t) =1+u 0 + u 1 , with γ =5andλ = 100. We note that the fronts pass through each other without generating spurious oscillations in the mesh. Note also that the solution points do not follow the front (moving in and out of the region of high mesh density), but that the density of the solution points does. Observe that the mesh automatically aligns itself along the front. A closeup of the mesh close to the front is given in Figure 3.11 (see p. 77). This shows very good resolution of the local structure close to the front and a smooth transition from a uniform mesh to one refined at the front. If γ is large, then close to the front we see a mesh compression proportional to γ orthogonal to the front, with a local skewness of s = γ. The PMA method in this case thus generates a smooth mesh, for which the degree of skewness can be controlled via the choice of γ. More details of this calculation are given in Walsh et al. (2009). We can also see the effects of mesh orthogonality close to the boundaries of the domain. Example 4. In this example we look at a monitor function localized on the support of two fronts meeting at an angle: see Figure 3.10. The behaviour of the mesh close to the front is similar to that in the last example. Significantly, there is no mesh tangling as the two fronts intersect. In Figure 3.12 we show a close-up of the intersection region, showing the high degree of mesh regularity. For this we take x 0 = t, y 0 =0.2+t/2, u 0 = γ sech(λ(x − x 0 + y − y 0 )), x 1 =1− t, y 1 =0.8 − t, u 1 = γ sech(λ(x − x 1 +(y − y 1 )/2)), M 4 (x, y, t) =1+u 0 + u 1 .
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Acta Numerica (2009), pp. 1-131 c
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