Radiation Transport Around Kerr Black Holes Jeremy David ...

146 CHAPTER 5. STEADY-STATE α-DISKS
nonlinear equations
f k (z n+1
k+1 , zn+1 k
, z n+1
k−1 ) =
− 1 ( z
n+1
k
− zk
n
∆t n ∆t n+1/2
− zn k − )
zn−1 k
+ pn+1 k+1/2 − pn+1 k−1/2
−
∆t n−1/2 Rẑˆtẑˆt
∆m (r)zn+1 k
= 0.(5.75)
k
On the right hand side, the pressure terms p n+1 are functions of the positions z n+1
k+1
and z n+1
k−1 . The function f k is well-defined by the equations above for the interior
zones (1 ≤ k ≤ N z − 1) and we use linear extrapolation to give f Nz , while planar
symmetry requires z−1 n+1 = −z1 n+1 , thus defining f 0 . The solution to equation (5.75)
gives the positions of the zone boundaries z n+1
k
, from which all the other hydrodynamic
variables can be determined.
Bowers & Wilson (1991) outline the standard approach to solving this set of
equations using Newton-Raphson iteration and a tridiagonal solver. Denoting the
first order solution to f k (t n+1 ) = 0 by the vector zk i , equation (5.75) can be written
as
f k (zk+1, i zk, i zk−1) i = f k (zk+1, n zk, n zk−1)
n
( ) n ( ) n ( ∂fk
+ ∆zk+1 n + ∂fk
∆zk n + ∂fk
∂z n k+1
∂z n k
Then an approximate solution to (5.75) is
∂z n k−1
) n
∆z n k−1 + O(∆z2 ). (5.76)
z n+1
k
= z n k + ∆z n k. (5.77)
We solve for these ∆zk n iteratively by setting f k(zk+1 i , zi k , zi k−1 ) = 0 and solving the
tridiagonal system
( ) n ( ) n ( ) n ∂fk
− ∆zk+1 n − ∂fk
∆zk n − ∂fk
∆zk−1 n = f k(zk+1 n , zn k , zn k−1 ) (5.78)
∂z n k+1
∂z n k
∂z n k−1
and then re-evaluating f k (zk+1 i , zi k , zi k−1 ) until an acceptable accuracy is reached for
z n+1
k
in equation (5.75). This typically take only about seven or eight iterations
to reach machine accuracy, due to the rapid convergence of the Newton-Raphson
root finding algorithm. This accuracy far exceeds the limiting first-order accuracy of
the finite difference equation for energy (5.71), so we generally use only four or five
iterations in the implicit scheme.
As mentioned above, the technique of operator splitting is employed to model the
energy transfer in the gas via radiation diffusion. Since the change in heat in a fluid
element is given by dQ, the rate of energy flow due to radiation and viscous heating