Adaptivity with moving grids

2 C. J. Budd, W. Huang and R. D. Russell
CONTENTS
1 Introduction 2
2 Moving mesh basics 10
3 Location-based moving mesh methods 44
4 Velocity-based moving mesh methods 83
5 Applications of moving mesh methods 91
References 121
1. Introduction
1.1. Motivation
Time-dependent systems of partial differential equations (PDEs) often have
structures that evolve significantly as the integration of the PDEs proceeds.
These can be interfaces, shocks, singularities, changes of phase, high vorticity
or regions of complexity. Associated with such structures are the
evolution of small length (and time) scales, rapid movement of the solution
features and the possibility of finite time blow-up of a component of the
solution. Frequently associated are also conservation laws, usually linked to
underlying symmetries. Examples of these phenomena occur in many applications,
such as gas and fluid dynamics, conservation laws, nonlinear optics,
free boundary problems, combustion, detonation, meteorology, mathematical
biology and nonlinear optics. To solve such PDEs numerically it is
typical to impose some form of spatial mesh and then to discretize the solution
on this mesh by using a finite element, finite volume, finite difference,
or collocation method. However, this strategy may not be effective in the
case of structures that involve small length scales, leading to large localized
errors. In such cases it is often beneficial to use some form of non-uniform
mesh, adapted to the solution, on which to perform all of the computations.
The advantages of doing this can be a reduced overall error, better conditioning
of the system, and better computational efficiency. Unfortunately,
introducing the extra level of complexity to the system through adaptivity
can also lead to additional computational cost and possible numerical instability.
Mesh adaptation should thus be used with care and appropriate
analysis where possible.
1.2. Adaptivity on a moving mesh
Adaptive methods for solving partial differential equations broadly fit into
three categories. The most extensively developed are static regridding methods,
in which a mesh is updated at each time level. The most widely used of
these are h-refinement methods, which form the basis of many commercial
codes. Usually such codes start with an initially uniform mesh, and then