Adaptivity with moving grids

Adaptivity with moving grids 115
Example 2. In closely related calculations we can compare with a similar
moving mesh calculation of the solution of Burgers’ equation, over the
interval x ∈ [0, 1], when ν =1e − 4 and with initial data u 0 = sin(2πx)+
(1/2) sin(πx). In this case we compute the mesh using a one-dimensional
form of PMA with the arclength monitor function and N = 30 mesh points,
central differencing in the computational domain for all spatial derivatives,
and solve the resulting ODEs using the MATLAB routine ode15s. Westart
with an initially uniform mesh, and evolve the mesh and solution together
using PMA with ɛ = 1. In this calculation, and with this choice of ɛ, the
mesh evolves from being uniform to one which is equidistributed, over a
time t ≈ 0.05. In this time period the underlying solution remains fairly
smooth. At the time t ≈ 0.2 the solution develops a sharp front, which is
well approximated, and then followed, by the evolving mesh. The solution is
presented in the physical domain at a series of different times in Figure 5.8,
and the resulting mesh trajectories are presented in Figure 5.9. Observe the
manner in which the mesh points resolve the front with no oscillations in
this case (due in part to the regularity of the initial data).
It is interesting to compare this calculation with a very similar calculation
made by Li and Petzold (1997), who looked at the solution of Burgers’
equation when ν =1e − 4 with the less regular initial data given by
u(x, 0) ≡ u 0 =0.2, x ≤ 0.1, u 0 =8x − 0.6, 0.1 ≤ x ≤ 0.2
u 0 =1, 0.2 ≤ x ≤ 0.5, u 0 = −10x +6, 0.5 ≤ x ≤ 0.6,
u 0 =0, 0.6 ≤ x ≤ 1.
This is a more severe test of the moving mesh method than the previous example,
as the initial data are much less smooth. This calculation employed
an arclength-based monitor function together with a regularized differential
algebraic formulation of the moving mesh equations, very similar to using
MMPDE6, and based on a method proposed in Adjerid and Flaherty (1986),
with the mesh-smoothing algorithm proposed by Dorfi and Drury (1987),
described in Section 2. Li and Petzold (1997) considered three different
discretizations: a central difference discretization, an ENO (Roe) method,
and a piecewise hyberbolic method (PHM) due to Marquina (1994). It was
found that in this case the central difference method tended to lead to spurious
oscillations due (as commented on in the example of the nonlinear
Schrödinger equation) to the destabilizing effect of the anti-diffusive terms
arising in the discretization of the advective terms describing the mesh motion.
Li and Petzold (1997) showed that these oscillations were reduced
with the ENO scheme and eliminated using the PHM method.
The related paper by Li et al. (1998) also considered applying the same
moving mesh method to a number of other reaction–diffusion problems,