Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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114 C. J. Budd, W. Huang and R. D. Russell<br />
1<br />
u<br />
1.4<br />
1.2<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
x<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0<br />
0<br />
0.2<br />
0.4<br />
0.6<br />
x<br />
0.8<br />
1 0<br />
0.2<br />
0.4<br />
y<br />
0.6<br />
0.8<br />
1<br />
0.2<br />
0.1<br />
0<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<br />
y<br />
Figure 5.7. Example 1. The solution of Burgers’ equation<br />
and the corresponding mesh at the time t =1.<br />
Table 5.1.<br />
N L 2 -error on a uniform mesh L 2 -error on the <strong>moving</strong> mesh<br />
20 6e-2 8e-3<br />
40 2e-2 2e-3<br />
80 6e-3 1e-3<br />
A quantitative measure of the error in the computed solution at t =1<br />
can be given by determining the L 2 -norm of the difference between it and<br />
the exact solution. We consider this error both for a uniform and a <strong>moving</strong><br />
mesh for various values of N: see Table 5.1.<br />
Sulman (2008) obtained a similar table by using a PMA method in which<br />
the convective terms are calculated by using an upwind method and the<br />
PMA system, and solved the underlying PDE alternately. It is also interesting<br />
to compare this calculation <strong>with</strong> the results presented in Zhang<br />
and Tang (2002). In this paper a harmonic mapping method of the form<br />
described in Section 4 was used to generate the mesh, and this was then<br />
coupled to a finite volume discretization of (5.31.) We see from Table 5.1<br />
that a useful reduction in the error for calculating the solution to (5.31)<br />
resulted from the PMA method. This reduction is similar to that observed<br />
in Zhang and Tang (2002) for the harmonic map method.