Adaptivity with moving grids

Adaptivity with moving grids 53
extensions to higher-order systems using higher-degree Hermite polynomials,
given in the code MOVCOL4 (Russell et al. 2007)) and this package
has been used in many tests of adaptive methods in one dimension: see,
for example, Huang and Russell (1996) and Budd et al. (1999a). Spline
collocation methods for second-order PDEs (see Ascher, Christiansen and
Russell (1981)) typically use a basis of third-degree cubic Hermite polynomials
to give a piecewise smooth approximation U(x, t) over a series of N
intervals x ∈ [X i (t),X i+1 (t)] to the solution u(x, t) of the underlying partial
differential equation and its associated boundary conditions The collocation
points are then chosen to be the Gauss points within the intervals. The interval
points are precisely the mesh points moved by solving the MMPDE.
The physical solution u(x, t) is approximated on the moving mesh by the
piecewise cubic Hermite polynomial
U(x, t) =U i (t)φ 1 (s (i) )+U x,i (t)H i (t)φ 2 (s (i) )
+ U i+1 (t)φ 3 (s (i) )+U x,i+1 (t)H i (t)φ 4 (s (i) ), (3.20)
for x ∈ [X i (t),X i+1 (t)], i=1, 2,..., N− 1, where U i (t) andU x,i (t) denote
the approximations to u(X i (t),t)andu x (X i (t),t), respectively. The local
coordinate s (i) is defined by
s (i) := (x − X i (t))/H i (t), H i (t) :=X i+1 (t) − X i (t), (3.21)
and the piecewise cubic shape functions are defined by
φ 1 (s) :=(1+2s)(1 − s) 2 , φ 2 (s) :=s(1 − s) 2 ,
φ 3 (s) :=(3− 2s)s 2 , φ 4 (s) :=(s − 1)s 2 .
(3.22)
For x ∈ [X i (t),X i+1 (t)], i =1,...,N − 1, we then have
U x (x, t) = 1 (
)
dφ 1
U i
H i ds + U dφ 2
x,iH i
ds + U dφ 3
i+1
ds + U dφ 4
x,i+1H i ,
ds
U xx (x, t) = 1 (
d 2 φ 1
Hi
2 U i
ds 2 + U d 2 φ 2
x,iH i
ds 2 + U d 2 φ 3
i+1
U t (x, t) = dU (
i
dt φ dUx,i
1 +
dt
+ dU i+1
φ 3 +
dt
( dXi
− U x (x, t)
dt
dH i
H i + U x,i
dt
(
dUx,i+1
dt
+ s (i) dH i
dt
)
φ 2
(3.23)
ds 2 + U d 2 )
φ 4
x,i+1H i
d 2 ,
s
(3.24)
)
φ 4
dH i
H i + U x,i+1
dt
)
, (3.25)