Adaptivity with moving grids

including the Fisher equation,
Adaptivity with moving grids 117
u t = βu xx + αu(1 − u). (5.34)
This equation has an exact solution with a moving front given by
u = (1 + exp( √ α/6x − 5αt/6)) −2 .
Whilst it was found that the wave front was well resolved, the moving mesh
method appeared to give worse results than for the fixed grid, with the front
moving at the wrong speed (too fast). Li et al. (1998) proposed that this
error was largely due to the effects of discretizing the additional convective
terms due to the moving mesh, as discussed in Section 3, which can lead
both to a high truncation error and instabilities in the solution. A possible
solution is to use higher-order methods or a curvature-dependent monitor
function; e.g., see Qiu and Sloan (1998). However, there is an interesting
alternative reason for the error. In problems such as (5.34), the speed of
motion of the front is not so much determined by the shape of the front
itself but by the nature of the exponential terms in the tails of the solution
and how they interact with the boundaries. Similar behaviour arises in
the Cahn–Hilliard equation, and also in similar systems such as the Gray–
Scott equation in chemistry. Somewhat paradoxically, the solution needs to
be well resolved in the boundary regions where it appears to be behaving
smoothly, in order for the front speed to be accurately resolved. Thus a
uniform mesh may well perform better than a moving mesh in these cases,
as it will probably be placing more mesh points close to the boundary.
5.4. Problems involving a change of phase and/or combustion
A very rich set of examples of the use of moving mesh methods is given
by Stefan-type problems, involving phase changes or combustion, and examples
of these can be found in Beckett et al. (2001a), Mackenzie and
Mekwi (2007a), Mackenzie and Robertson (2002), Miller, Gleyzer and Imhoff
(1998), Huang and Zhan (2004), Zegeling (2005), Tan (2007) and Tan
et al. (2007).
As an example we consider the combustion problem described in Cao
et al. (1999a) and Moore and Flaherty (1992). The mathematical model is
a system of coupled nonlinear reaction–diffusion equations,
∂u
∂t − ∇2 u = − R αδ u eδ(1−1/T ) ,
∂T
∂t − 1 Le ∇2 T =
R
δLe u eδ(1−1/T ) ,
where u and T represent the dimensionless concentration and temperature
of a chemical which is undertaking a one-step reaction. We consider the
J-shape solution domain shown in Figure 5.10. The initial and boundary