Adaptivity with moving grids

Adaptivity with moving grids 79
properties of the solution such as scaling structures (Budd et al. 1996, Budd
and Williams 2006, Baines et al. 2006). Alternatively, the moving mesh
equations and (3.49) can be solved alternately (Huang 2007, Huang and
Russell 1999, Ceniceros and Hou 2001). This reduces the stiffness problems
but can lead to a lag in the mesh movement.
Alternatively, a rezoning method can be used (Tang 2005). In such methods,
the MMPDEs are solved to advance the mesh by one time step. The
current solution u is then interpolated onto this new mesh in the physical
domain, and the original PDE (3.48) solved on the new mesh in this domain
(often using a finite element or finite volume method). The advantage of
this method is that standard software can be used to solve the PDE, but
at the expense of an interpolation step. We now describe these methods in
more detail.
3.4.1. Simultaneous solution in the computational domain
We first briefly detail the implementation of the simultaneous solution
method solving the MMPDEs and (3.49) by using finite differences in the
computational domain. To do this we associate each mesh point in the fixed
computational mesh with a solution point U i,j (t). To discretize (3.49) we
transform the derivatives in the physical domain to ones in the computational
domain. For example, in two dimensions, if we have general transformation
F with Jacobian J then the derivatives of u can be expressed in
terms of the computational variables in the following manner:
u x = 1 J
(
yη u ξ − y ξ u η
)
, (3.50)
u y = 1 ( )
−xη u ξ + x ξ u η ,
J
u xx = 1 ( (
yη J −1 )
y η u ξ
J
ξ − y (
η J −1 )
y ξ u η ξ − y (
ξ J −1 )
y η u ξ η + y (
ξ J −1 )
y ξ u η η)
,
u yy = 1 ( (x η J −1 )
x η u ξ
J
ξ − x (
η J −1 )
x ξ u η ξ − x (
ξ J −1 )
x η u ξ η + x (
ξ J −1 ) )
x ξ u η .
η
As the computational domain has a uniform mesh, and the Jacobian J may
be determined directly from the mesh mapping F , it follows that (3.50) can
be discretized to high accuracy in space using a standard finite difference or
finite element discretization. The solution u is then determined by solving
(3.46) and (3.49) together using an appropriate ODE solver such as SDIRK
or a predictor–corrector method. Examples of the use of this method are
given in Section 5.
3.4.2. Alternating solution in the computational domain
The alternating solution method is often used in higher-dimensional calculations
to alternatively solve the system and to move the mesh. This method