Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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<strong>Adaptivity</strong> <strong>with</strong> <strong>moving</strong> <strong>grids</strong> 91<br />
attractive and repulsive pseudo-forces between nodes, motivated by a spring<br />
model in mechanics. Petzold (1987) obtains an equation for mesh velocity by<br />
minimizing the time variation of both the unknown variable and the spatial<br />
coordinate in computational coordinates and adding a diffusion-like term to<br />
the mesh equation. The method of Hyman and Larrouturou (1986, 1989)<br />
also simulates attraction and repulsion pseudo-forces.<br />
In contrast, Dorodnitsyn (1991, 1993a, 1993b) and his co-workers (Dorodnitsyn<br />
and Kozlov 1997, Kozlov 2000) have derived a series of velocitybased<br />
methods in which the underlying symmetries of the PDE to be solved<br />
are used to guide the movement of the grid points. Such methods have<br />
the potential to preserve all of the symmetric invariants of the underlying<br />
PDE in the discretized system, and indeed precisely these invariants are<br />
used in the calculation of the mesh points. They are closely related to the<br />
(position-based) scale-invariant methods described in the next section. Such<br />
methods can also lead to conservation laws derived from a discrete version<br />
of Noether’s theorem. They have been applied in Dorodnitsyn and Kozlov<br />
(1997) to solve a variety of nonlinear diffusion equations, and in Budd and<br />
Dorodnitsyn (2001) to integrate the nonlinear Schrödinger equation. The<br />
main disadvantage of these methods is that they tend to be highly nonlinear,<br />
and the equations for the mesh points hard to solve. More details of the<br />
implementation and application of these methods are given in Budd and<br />
Piggott (2005).<br />
5. Applications of <strong>moving</strong> mesh methods<br />
In this final section we look at some applications of <strong>moving</strong> mesh methods to<br />
a variety of problems related to the solution of partial differential equations.<br />
As described in Section 1, there are a vast number of problems for which<br />
<strong>moving</strong> mesh methods have been used <strong>with</strong> great success, and we cannot<br />
hope to summarize all of them in this section. For example, one of the<br />
most important applications arises in fluid mechanics, and reviews of these,<br />
comparing many different methods and problems, are given in Baines (1994),<br />
Eisman (1985), Tang (2005), Tang and Tang (2003) and Yanenko et al.<br />
(1976).<br />
Instead, we look in this section at some specific problems and classes of<br />
problems which aim to highlight some of the special advantages and disadvantages<br />
of the <strong>moving</strong> mesh method for solving partial differential equations.<br />
In particular we look at problems where <strong>moving</strong> meshes can exploit<br />
natural solution scaling structures (including self-similarity); blow-up problems<br />
in which solution singularities arise in finite time; and problems such as<br />
Burgers’ equation, where <strong>moving</strong> meshes are used to capture the motion of<br />
<strong>moving</strong> fronts, and two physical problems, one in combustion and the other<br />
in meteorology. Two primary goals are to describe analytical techniques for