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