539 / 1228
The LBTE
)
(
4
)ˆ, ,(
)ˆ, ,(
ˆ
pr r
P pq
scat q
Er t
Er
There is no analytical (closed form) solution of the LBTE.
1
p
scat
This is because it is an equation of 6 variables which is integro-differential
since the scatter source is:
Ed d Er
E Ers
Er scat
q
ˆ )ˆ, ,( )ˆ ˆ,
,(
04
)ˆ, ,(
i.e. : I must integrate for the scatter source and this depends
on the solution itself at all other (E’,Ω’) points of the phase space…!
h l
i i l h
i
i ll
T e on y opt on s to so ve t e equat on numer ca y
(and this is exactly what Acuros and similar algorithms do) by:
separating variables:
)ˆ( ) ( )(
)ˆ( ) , (
)ˆ, ,(
YEf r
YEr
Er
solve the LBTE iteratively:
•
make initial guess for
•
approximate integration for the scatter source by summation over discrete E, Ω
)(
r
elements
•
approximate derivatives by finite differences over discrete space elements
•
correct initial guess and continue until a convergence criterion is met
,