Adaptivity with moving grids

8 C. J. Budd, W. Huang and R. D. Russell
been developed to solve this problem such as a variational method, the
geometric conservation law, moving mesh PDEs and optimal transport
methods.
(3) The underlying PDE is then discretized, either on the mesh in the
computational domain or in the original physical domain (in the latter
case a finite element or finite volume method is usually employed
(Tang 2005)). The underlying partial differential equation and the
mesh equations can then be solved either simultaneously, typically
by using the method of lines (Huang, Ren and Russell 1994), or alternatively
(often by using a predictor–corrector method). The first
method avoids the need for any interpolation from one mesh to another,
but is usually associated with having to solve stiff differential
equations. Alternating solutions can be implemented using either the
quasi-Lagrange approach (Huang and Russell 1997b) or the rezoning
approach (Tang 2005). The former transforms time derivatives to those
along mesh trajectories and avoids interpolation of the physical solution
from the old mesh to the new one. However, it has the disadvantages
that it has to deal with extra convection terms caused by mesh
movement and may cause a time lag in mesh movement. On the other
hand, the rezoning approach solves the physical PDE on a fixed mesh
over a time step but requires interpolation from one mesh to another
(which often has to be done very carefully to preserve conservation
laws). We will consider both methods in detail in this article.
We are currently in a situation where the mesh formulation problem, mesh
generation and the solution of PDEs on a moving mesh are generally well
understood in one spatial dimension. Reliable and efficient moving mesh
methods exist (and are implemented in a number of packages) which are
based on such formulations and can be used to solve time-evolving PDEs
in one spatial dimension, with associated error estimates in certain cases.
Indeed, for such problems the use of moving mesh PDEs to evolve the mesh
coupled with a method of lines approach has proved to be very effective, and
also amenable to analysis. In this article we will be able to give a detailed
description of the theory, implementation and application of such methods.
However, the problem of mesh movement, and the discretization of PDEs
on such meshes, is much less understood in higher dimensions, and this will
form the bulk of the discussion in this paper.
1.4. A historical survey
Moving meshes and the use of adaptive strategies to minimize estimates of
the solution error have a rich and diverse literature. Moving mesh methods
can be classified according to the mesh movement strategy into two groups
(Cao, Huang and Russell 2003): velocity-based methods and location-based