Adaptivity with moving grids

Adaptivity with moving grids 45
function of x. It immediately follows that x is a unique monotone increasing
function of ξ. Observe further that this function is as smooth as the
function M.
This simple observation makes equidistribution relatively easy in one dimension.
The most direct way to enforce equidistribution is to solve (3.1) directly.
However, this has the disadvantage that it requires the calculation of the
integral θ. This can be avoided by a further differentiation with respect
to ξ, and thus solving the moving mesh equation (together with boundary
conditions) given by
(Mx ξ ) ξ =0, x(0,t)=a, x(1,t)=b. (3.3)
To determine an equidistributed mesh, the equation (3.3) can be discretized
over the computational domain and then solved. This discretization
does not have be done to high accuracy in order to obtain a regular
mesh suitable for solving the underlying PDE. A typical such discretization
takes the form
E i ≡ 2 (
Mi+1/2
∆ξ 2 (X i+1 − X i ) − M i−1/2 (X i − X i−1 ) ) =0,
M i+1/2 = 1 2 (M i + M i+1 ). (3.4)
However, the solution of the system (3.4) requires solving a system of
nonlinear equations, which is usually difficult and requires the use of some
form of iterative procedure. See Pryce (1989), Xu, Huang, Russell and
Williams (2009), He and Huang (2009), Kopteva and Stynes (2001) and
Kopteva (2007) for a discussion of such methods, and conditions for them
to converge to a solution.
This problem can be avoided by instead introducing a natural time evolution
into the mesh equations. Differentiating the equidistribution equation
(3.3) with respect to time gives the (so-called ) MMPDE0 (Huang
et al. 1994):
d ( )
(Mxξ ) ξ =0. (3.5)
dt
Instead we may also differentiate (3.1) with respect to time (Adjerid and
Flaherty 1986). This leads directly to the GCL method described in the
next section. The resulting equation then takes the form
∂
∂ξ (Mx t)+M t x ξ = θ t . (3.6)
This equation can then also be differentiated with respect to ξ to eliminate
the θ contribution, giving MMPDE1 (Huang et al. 1994), where it is assumed
that we can find M t , although in practice this may not be easy.