Adaptivity with moving grids

78 C. J. Budd, W. Huang and R. D. Russell
Further examples in which we couple the PMA algorithm to generate
moving meshes for certain partial differential equations are presented in
Section 5.
3.4. Adaptive discretization of PDEs in higher spatial dimensions
We now extend the discussion earlier in this section, in which we looked
at coupling an MMPDE to a one-dimensional PDE, to consider the harder
problem of coupling a moving mesh method (derived either from a variational
approach or from a Monge–Ampère-based approach) to a partial
differential equation in several spatial dimensions, which we assume has
the form
u t = f(t, x, u, ∇u, ∆u). (3.48)
Obviously, in this case the errors in using a non-uniform mesh are more
pronounced, and the additional convective terms introduced by the moving
mesh are potentially destabilizing. A moving mesh method becomes
an adaptive method when it is coupled to a discretization of a partial differential
equation. There is no unique way to perform this coupling, and
it depends upon whether the PDE is discretized in the computational domain
(typically with a finite difference method) or in the physical domain
(typically with a finite element or a finite volume method). The nature
of the coupling also depends upon whether or not the adaptive method is
going to be coupled to existing software (such as a standard CFD method).
The former usually involves some form of interpolation of the solution to
map it onto a mesh suitable for the existing solver to use. The coupling of
the mesh to the underlying PDE should also preserve important structures
of the PDE; in particular, any conservation laws or solution symmetries
should ideally be preserved in the coupled system. This can be done in
various ways: see Tang (2005), Huang (2007) for reviews of these. The
quasi-Lagrangian approaches (which avoid interpolation) allow directly for
mesh movement, and express the PDE in Lagrangian variables, generalizing
the expression (3.16):
˙u = f(t, x, u, ∇ x u, ∆ x u)+∇ x u · ẋ, (3.49)
where ˙u, ẋ denote derivatives with respect to time, with the computational
variable ξ fixed. The MMPDEs and (3.49) can then both be discretized
on the computational mesh and solved simultaneously. As described earlier,
this procedure is generally the method of choice when used in calculations
in one spatial dimension (Huang and Russell 1996). However, it is much
harder to use in higher dimensions as the coupled system can be very stiff
and the equations are very nonlinear. This method has the advantages
that there is no need to interpolate a solution from one mesh to the next
as the mesh evolves, and also that the mesh can inherit useful dynamical