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