Adaptivity with moving grids

24 C. J. Budd, W. Huang and R. D. Russell
It follows from a change of variable that
∫
∫ A dξ
∫
Ω C
dξ = A
M(x(ξ,t),t)|J(ξ,t)| dξ
∫
. (2.17)
Ω P
M(x(ξ,t),t)dx
As the set A is arbitrary, the map F(ξ,t) must (for all (ξ,t)) obey the
identity
∫
Ω
M(x(ξ,t),t)|J(ξ,t)| = θ(t), where θ(t) = P
M(X(ξ,t),t)dx
∫
. (2.18)
dξ
We shall refer to (2.18) as the equidistribution equation. This equation must
always be satisfied by the map F(ξ,t). It is the central equation of much of
mesh generation, and we shall show presently that it is strongly connected
to a variational representation of the mesh transformation.
2.5.2. Choice of a scalar monitor function
The choice of a scalar monitor function M appropriate to the accurate
solution of a PDE is difficult, problem-dependent, and the subject of much
research. We do not consider this in detail here but give a brief review of
various choices used for certain problem classes. The function M can be
determined by aprioriconsiderations of the geometry or of the physics of
the solution. An example is the generalized solution arclength given by
M = √ 1+c 2 |∇ x u(x)| 2 , or alternatively M =
Ω C
√
1+c 2 |∇ ξ u(x(ξ))| 2 .
(2.19)
The first of these is often used to construct meshes which can follow moving
fronts with locally high gradients (Winslow 1967, Huang 2007). A careful
analysis of the application of arclength-based monitor functions to the resolution
of the solution of singularly perturbed PDEs is given in Kopteva
and Stynes (2001). Ceniceros and Hou (2001) successfully used the second
monitor function (with u being the temperature) to resolve small scale singular
structures in Boussinesq convection. It is also common to use monitor
functions based on the (potential) vorticity, or curvature, of the solution
(Beckett and Mackenzie 2000), and these have been used in computations
of weather front formation (Budd and Piggott 2005, Budd, Piggott and
Williams 2009, Walsh, Budd and Williams 2009). In certain problems,
moving fronts are associated with changes in the physics of the solution.
An example is problems with phase changes, where the phase front occurs
at those points (x m ) i at a temperature T = T m . Insuchcasesitis
possible to construct meshes which resolve behaviour close to the phase
boundary by using the monitor function M = a/ √ b|x − x m | + c, where
|x − x m | =min|x − (x m ) i | (Mackenzie and Mekwi 2007a). Alternatively, M
can be linked to estimates of the solution error. A significant calculation in