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