Adaptivity with moving grids

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