Adaptivity with moving grids

48 C. J. Budd, W. Huang and R. D. Russell
These can then be solved using standard stiff ODE software, e.g., byusing
an SDIRK (singly diagonally implicit Runge–Kutta) method. A simple such
semi-discretization of (3.9) is given by
(
)
ɛ Ẋ i − γ Ẋi+1 − 2Ẋi + Ẋi−1
(∆ξ) 2 = E i (t), (3.11)
where the equidistribution measure E i (t) is as given in (3.4). This leads
(on inversion of the simple tri-diagonal system on the right-hand side of
this equation) to a simple set of ODEs for the location of the mesh points.
Alternatively, a simple full discretization of (3.9) for a mesh Xi
n evaluated
at the time t n = n∆t is proposed in Ceniceros and Hou (2001), and takes
the form
(
ɛ
X n+1
i
(
ɛ
− γ Xn+1 i+1 − 2Xn+1 i
+ Xi−1
n+1 )
(∆ξ) 2 =
Xi n − γ Xn i+1 − 2Xn i + Xi−1
n )
(∆ξ) 2 +∆tEi n . (3.12)
These equations for the mesh can then be solved together with a suitable
discretization of the underlying PDE, either simultaneously or alternately.
We will give more details of this procedure later in this section in the context
of moving meshes in higher dimensions, but we note at this stage that the
simultaneous solution method is both possible and effective in such onedimensional
problems.
The method for evolving the mesh is typically implemented in two stages.
(1) Starting from an initially uniform mesh in the physical space, we evolve
this to equidistribute the monitor function at the initial time over Ω P .
To do this we set M 0 (x) ≡ M(x, 0), with x ξ = a +(b − a)ξ, and
solve (3.46) with M fixed to equal the function M 0 ,withɛ =1for
0 < t < T, where T is a fixed time. In this first calculation t is
an artificial time during which the uniform mesh evolves toward an
equidistributed mesh for which the right-hand side of (3.9) is zero.
It follows from the earlier results that, provided M 0 > 0, such a mesh
exists, and we show presently that it is stable. In this initial calculation
the right-hand side of (3.9) is initially relatively large, and taking ɛ =1
prevents the numerical calculation of the solution of the ODEs for
the mesh point locations from being unnecessarily stiff. The value
of T is chosen large enough to allow the mesh to relax toward the
equidistributed state.
(2) We then solve (3.9) with the true time-dependent monitor function
M(x, t), with t now actual time. For this calculation we typically set
ɛ =0.01. We show presently that the resulting mesh is then ɛ-close to a
mesh which exactly equidistributes M(x, t), provided that M does not