Adaptivity with moving grids

4 C. J. Budd, W. Huang and R. D. Russell
as the solution evolves. Hence the use of the alternative name of moving
mesh methods for such procedures. The mesh points are then concentrated
into regions where the solution has ‘interesting behaviour’, usually typified
by a rapid variation of either the solution or one of its (higher) derivatives.
The objective of this approach is to get the smallest error possible for the
number N of mesh points used, and to try to obtain error estimates which
depend upon the value of N but not on the solution itself. For example,
if the solution evolves a boundary layer of width ɛ (with ɛ decreasing as
time advances), then ideally the mesh points should concentrate into this
boundary layer so that the solution error is independent of ɛ. The moving
mesh methods typically work by generating a mapping from a regular
(logical or computational) domain into a physical domain in which the underlying
equation is posed. The location, or the velocity, of the mesh points
is then determined by solving a system of auxiliary partial differential equations,
often called the moving mesh equations. In all cases a vector or a
scalar monitor function (or functions) is used to guide the position of the
mesh. The monitor function is usually designed to give an estimate of some
measure of the solution error which is then equidistributed over each mesh
cell. The monitor function is usually constructed in one of three ways. It
may depend upon a priori solution estimates (such as arclength or curvature),
on a posteriori error estimates (such as the solution residual, as
used in moving finite element methods (Baines 1994), or estimates of the
derivative jump across element boundaries (Tang 2005)), or on some underlying
physics related to the solution, such as the potential temperature
or the vorticity in a meteorological problem (Budd and Piggott 2005). In
the case of scale-invariant problems, such physical estimates are often optimal.
When using such methods, much care has to be taken in preventing
mesh tangling and ensuring mesh regularity and isotropy (where relevant).
A discussion of this will form a significant part of Section 2. We also require
that discretizations of the underlying PDE on such meshes (in either
the computational or the physical domain) should retain important properties
of the underlying physical solution, such as conservation laws and
scaling structures (Tang 2005). Provided that these conditions are carefully
considered, r-adaptive methods can be used with considerable success for
many time-evolving systems. Examples of these include computational fluid
dynamics (Yanenko, Kroshko, Liseikin, Fomin, Shapeev and Shitov 1976),
groundwater flow (Huang and Zhan 2004, Huang, Zheng and Zhan 2002),
blow-up problems (Budd, Huang and Russell 1996, Budd and Williams 2006,
Ceniceros and Hou 2001, Ren and Wang 2000), chemotaxis systems (Budd,
Carretero-Gonzalez and Russell 2005), reaction–diffusion systems (Zegeling
and Kok 2004), the nonlinear Schrödinger equation (Sulem and Sulem 1999,
Ceniceros 2002, Budd, Chen and Russell 1999a), phase change problems
(Mackenzie and Robertson 2002, Mackenzie and Mekwi 2007a, Tan,Lim