Adaptivity with moving grids

10 C. J. Budd, W. Huang and R. D. Russell
and Hansen (1991), Thompson (1985), Thompson, Warsi and Mastin (1982)
and Thompson and Weatherill (1992), and the books of Baines (1994),
Carey (1997), Knupp and Steinberg (1994), Liseikin (1999), Thompson,
Warsi and Mastin (1985) and Zegeling (1993). In particular, Hawken et al.
(1991) give an extensive overview and references on moving mesh methods
before 1990. In addition to the references cited above, we would also
like to bring the reader’s attention to the recent interesting work of Bank
and Smith (1997), Beckett, Mackenzie and Robertson (2001a), Budd et al.
(1996), Calhoun, Helzel and LeVeque (2008), Ceniceros and Hou (2001),
Chacón and Lapenta (2006), Lapenta and Chacón (2006), Di, Li, Tang and
Zhang (2005), Huang and Zhan (2004), Mackenzie and Robertson (2002),
Ren and Wang (2000), Stockie et al. (2000), Tang and Tang (2003) and
Zegeling and Kok (2004) on moving mesh methods and their applications.
1.5. Outline of this article
The purpose of this Introduction has been to give an underlying motivation
for the theory and application of (adaptive) moving meshes. In Section 2
we will consider in detail the geometry of possible meshes (with special
regard to equidistribution and isotropy), and the nature of discretizations of
differential equations on them. In Section 3 we then look in detail, and with
reference to many examples, at ‘location-based’ meshes in which the local
density of the mesh points is controlled by a monitor function. These include
moving mesh PDE (MMPDE) methods, variational methods and optimaltransport-based
methods. This discussion will look at moving meshes in
both one and higher dimensions and compare the strategies used for these
two cases. In Section 4 we will then look at velocity-based methods, such as
the geometric conservation law (GCL) methods and the moving mesh finite
element methods, in which the velocity of the mesh points, rather than their
position, is controlled. The concluding section, Section 5, will then look at
some examples in much more detail, considering scale invariance, blow-up
problems, problems with convection and moving fronts, phase change and
combustion problems, and problems arising in meteorology.
2. Moving mesh basics
In this section we will give an overview of the main aspects of adaptive
moving mesh generation, and will concentrate on the nature of the geometry
of an adapted mesh, the equidistribution and variational approaches to
defining a mesh, and the relation of the mesh to solution (truncation and
interpolation) errors. The movement of the mesh and the way that it can
be coupled to a partial differential equation will be discussed briefly, but will
mainly be the subject of Sections 3 and 4.