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