Adaptivity with moving grids

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