Adaptivity with moving grids

Adaptivity with moving grids 51
mesh equations and the PDE equations can then be solved together or alternately.
We will discuss this in more detail presently in the context of
moving mesh methods in higher dimensions, but it is appropriate to make
some preliminary remarks here.
3.1.2. Finite difference methods
To motivate the discussion of appropriate discretizations, we assume that
the underlying PDE system takes the form
u t = f(t, x, u, u x , u xx ). (3.15)
If x(ξ,t) is itself a time-dependent function of a computational variable ξ
then (3.15) can be cast into the Lagrangian form in the moving coordinate
system given by
du
dt = f(t, x, u, u x, u xx )+u x x t . (3.16)
The MMPDE governing the mesh motion gives a direct value for x t . A
method effective for solving (3.16) (in one-dimensional problems) is to use a
semi-discretization. In this approach we discretize the differential equation
(3.16) in the computational coordinates together with a similar discretization
of the MMPDE (3.9). In such a semi-discretization we set
X i (t) ≈ x(i∆ξ,t) and U i (t) ≈ u(X i (t),t).
As a simple example of the use of a finite difference method we can then
take
u x ≈ U i+1 − U i−1
X i+1 − X i−1
and u xx ≈
U i+1 −U i
X i+1 −X i
− U i−U i−1
X i −X i−1
. (3.17)
X i+1 −X i−1
2
These discretizations can then be substituted into (3.16) and the resulting
set of ODEs for X i and U i solved along with one of the discretizations of
(3.9). We discuss presently, and in more detail, the various alternating
and simultaneous approaches for discretizing in time and then solving the
resulting combined system.
On a static mesh, the truncation errors in calculating these finite difference
approximations were given in the expressions (2.49) and (2.52). Provided
that the stretching condition (2.10) is satisfied, then these errors are
of second order.
We note, however, that additional errors may arise from the additional
convective terms arising from the mesh movement, in particular the term
u x x t (3.18)
arising in (3.16). This additional term leads to both theoretical and practical
difficulties in applying the moving mesh methods. From a theoretical