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