Multilevel transport solution of LWR reactor cores Page: 4 of 8

                                
routine is called. These diagonal blocks can be
interpreted as the coefficient matrix for each
moment of the even parity transport equation.
2.2 Multigrid algorithms
The proposed method uses a multigrid algo-
rithm to solve the in-moment equation. The
basis for multigrid methods is to take advantage
of the spectral properties of stationary iterations,
i.e. the smoothing effect of these iterations on
the error (Fig. 1).
~~c
(a)
(b)
JCo
(c)
Fig. 1: Effect of (a) one, (b) 10, and (c) 99 point-
Jacobi iterations on the error of a homogeneous
problem with a random guess.
After a few iterations the change in the error is
negligible and the efficiency of the method based
on stationary iterations decreases. At this point,
multigrid methods compute the residual and pro-
ject it onto a coarser grid using a restriction

operator. In this new grid, an additive correction
is sought by solving the error equation:

Ac e=Rfr

(12)

where Rfr is the projected residual. In the
coarser mesh the error is represented by higher
frequency modes, which are attenuated with
higher efficiency by the stationary iterations.
After obtaining this additive correction, the
solutions is projected back into the fine mesh
using an interpolation operator, and then applied
to the solution.
:1S +1- xi + 12'e
xf9 - xIsec
If this method is applied over a set of n meshes
each one coarser than the previous, we arrive to
the algorithm for a multigrid V-cycle:
- Relax Aox = b, vi times (vl is usually 1)
using the initial value x(o) as guess.
-   Compute the residual: ro = b - Aox("1.
- Restrict the residual into the next coarse
grid: bi= Roro.
- Relax Alx1 = bl, vi times using
e() as initial value.
-   Compute the residual: r"1 = b     -
Aix "v).
- Restrict the residual into the next
coarse grid: b2= Rir.
-   Solve Ax, = b,.
- Interpolate the error and correct
the solution: x1= x1 +1x2.
- Relax Alx1 = bl, v2 times (v2 is
usually 1).
- Interpolate the error and correct the solu-
tion: xo=xol+11x1.
- Relax Aoxo = bo, v2 times
- Check the convergence.
3. Implementation
The implementation of the multigrid algorithm
required the modification of the EVENT solver

3