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There is no analytical (closed form) solution of the LBTE.
This is because it is an equation of 6 variables which is integro-differential
since the scatter source is:
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…!
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The LBTE
The only option is to solve the equation numerically
(and this is exactly what Acuros and similar algorithms do) by:
➢
separating variables:
➢
solve the LBTE iteratively:
•
make initial guess for
•
approximate integration for the scatter source by summation over discrete E, Ω
elements
•
approximate derivatives by finite differences over discrete space elements
•
correct initial guess, and continue until a convergence criterion is met
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