Adaptivity with moving grids

82 C. J. Budd, W. Huang and R. D. Russell
A one-dimensional calculation. Suppose that the ith new grid point at time
t n+1 is Xi n+1 . In a finite volume method the key quantities are the cell
averages given by
∫
1 Xj+1
U j+1/2 =
u dx, X
X j+1 − X j+1/2 = 1
j X j
2 (X j + X j+1 ).
If these are known on the mesh x n then they can be interpolated onto
the new mesh x n+1 . Suppose that the new mesh points satisfy X n+1
j+1/2 ∈
[Xk−1/2 n ,Xn k+1/2
]; then, naively, this can be done via the formula
Ûj+1/2 n = U k+1/2 n + U k+1/2 n − U k−1/2
n
Xk+1/2 n − (X n+1
Xn j+1/2 − Xn+1 k+1/2 ).
k−1/2
Unfortunately, this simple linear interpolation does not conserve discrete
solution mass, in the sense that
∑ (
Û
n
j+1/2
X
n+1
j+1 − ) ∑ ( Xn+1 j ≠ U
n
j+1/2
X
n
j+1 − Xj
n )
,
and consequently leads to unsatisfactory results (Tang and Tang 2003) when
used to solve hyperbolic conservation laws. An improved method in Tang
and Tang (2003) mimics in part the discussion above for the relation between
the quasi-Lagrangian method and the rezoning method. Suppose
that x n+1 = x n + δt ẋ; then it follows from an application of the Reynolds
transport theorem that, to leading order,
(X n+1
j+1 − Xn+1
j
)Û n j+1/2 ≈ ∫ X
n+1
j+1
X n+1
j
û n dx
≈ (X n j+1 − X n j )U n j+1/2 +∆t((ẋU n ) j+1 − (ẋU n ) j ).
This prompts the use of the conservative (to leading order) interpolation
formula given by
(Xj+1 n+1 − Xn+1 j
)Û j+1/2 n =(Xn j+1 − Xj n )u n j+1/2 +∆t((ẋU n ) j+1 − (ẋU n ) j ),
(3.51)
which automatically conserves discrete mass. Tang and Tang (2003) use
this method with success to solve conservation laws of the form
u t + f(u) x =0
with the PDE being integrated using a MUSCL method (LeVeque 1990)
and using the MMPDE (3.9) to advance the mesh.
Two-dimensional calculations. In two dimensions, Tang and Tang (2003)
use the moving mesh method described in Ceniceros and Hou (2001) to advance
the mesh (X, Y ) from time t n to time t n+1 using the MMPDE (3.34).