Adaptivity with moving grids

In this calculation we suppose that
Adaptivity with moving grids 83
(x n+1 ,y n+1 )=(x n ,y n )+∆t(ẋ, ẏ).
Cell averages at time t n over a (time-evolving mesh cell) of area A n j+1/2,k+1/2
are now given by Uj+1/2,k+1/2 n , etc., and the normal mesh speed relative to
the surfaces of the mesh cell is given by
v =ẋn x +ẏn y , where the unit normal is given by (n x ,n y ).
The conservative interpolation scheme proposed is
A n+1
j+1/2,k+1/2Û j+1/2,k+1/2 n = An j+1/2,k+1/2 U j+1/2,k+1/2 n (3.52)
+∆t ([ (vU n ) j+1,k+1/2 +(vU n ]
) j,k+1/2
+ [ (vU n ) j+1/2,k+1 +(vU n ) j+1/2,k
])
.
This scheme preserves discrete mass to leading order. Again a MUSCL
method can be used to advance the solution of the PDE. This method is
then used to solve the double-Mach reflection problem and various other
problems with contact discontinuities arising in the solution of the Euler
equations.
4. Velocity-based moving mesh methods
In this section we will look in some more detail at velocity-based methods
for moving meshes. These are also called Lagrangian methods, and they
rely on calculating the mesh point velocities and from this the mesh point
locations. In some ways these methods are very natural, since in (say) fluid
mechanics calculations, natural solution features are often convected with
the flow, and it is natural to evolve the mesh points to follow the flow itself.
(Note the huge popularity of the semi-Lagrangian and the characteristic
Galerkin methods.) However, velocity-based methods can easily have severe
implementation problems, and overcoming them remains a challenging
issue. These include significant mesh tangling, with associated skewness,
and also a tendency to create meshes which lag behind the solution. Indeed,
it is very possible for such meshes to have unstable movement, to
move well away from equidistributed solutions and to lead to permanently
distorted skewed and even frozen meshes, even after the significant solution
structures have long gone. Some examples of this type of behaviour will
be presented in our discussion of the GCL method. For these reasons the
overall performance of these methods is, in general, not as good as that
of the position-based methods described in the last section, and hence we
will spend less time discussing them. We now describe three velocity-based
methods: the moving finite element method (MFE), the geometric conservation
law method (GCL), and the deformation map method.