In this paper the spherical harmonic function method for the plane geometry neutron transport equation with reflective boundary condition is discussed. The existence and uniqueness for the solution of the spherical harmonic approximation (PN) equation are studied, and then the convergence of the solution of the PN equation as N→∞ to the solution of the neutron transport equation is proved.

1.
G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).
2.
K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, MA, 1967).
3.
S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
4.
R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Method in Science and Technology, Vol. 6 (Springer Verlag, Berlin, 1993).
5.
J. A.
Davis
, “
Transport error bounds via PN approximations
,”
Nucl. Sci. Eng.
31
,
127
146
(
1968
).
6.
B.
Davison
, “
On the rate of convergence of the spherical harmonics method
,”
Can. J. Phys.
38
,
1526
1545
(
1960
).
7.
J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley-Interscience, New York, 1979).
8.
E. Lewis and W. Miller, Computational Methods of Neutron Transport (Wiley, New York, 1984).
9.
R. D. Richtmyer and K. W. Morton, Difference Methods for the Initial Value Problem, 2nd ed. (Interscience, New York, London, 1967).
10.
E.
Larsen
, “
Diffusion theory as an asymptotic limit of transport theory for nealy critical systems with small mean free paths
,”
Ann. Nucl. Energy
7
,
249
255
(
1980
).
11.
E.
Larsen
and
J.
Keller
, “
Asymptotic solution of neutron transport problems for small mean free paths
,”
J. Math. Phys.
15
,
75
81
(
1974
).
12.
R. P.
Rulko
, “
Variational derivation of multigroup P2 equations and boundary conditions in planar geometry
,”
Nucl. Sci. Eng.
121
,
371
392
(
1995
).
13.
R. P.
Rulko
and
E. W.
Larsen
, “
Variational derivation and numerical analysis of P2 theory in planar geometry
,”
Nucl. Sci. Eng.
114
,
271
285
(
1993
).
14.
R. P.
Rulko
,
D.
Tomašević
, and
E. W.
Larsen
, “
Variational P1 approximations of general geomerty multigroup transport problems
,”
Nucl. Sci. Eng.
121
,
393
404
(
1995
).
15.
C.
Bardos
,
R.
Santos
, and
R.
Sentis
, “
Diffusion approximation and computation of the critical size
,”
Trans. Am. Math. Soc.
284
,
617
649
(
1984
).
16.
M.
Ribarič
and
L.
Šušteršič
, “
Asymptotic approximations to a linear transport equation
,”
Transp. Theory Stat. Phys.
23
,
815
843
(
1994
).
17.
E.
Larsen
and
G. C.
Pomaraning
, “
The PN theory as an asymptotic limit of transport theory in planar geometry-I: analysis
,”
Nucl. Sci. Eng.
109
,
49
75
(
1991
).
18.
K. O.
Friedrichs
, “
Symmetric positive linear differential equations
,”
Commun. Pure Appl. Math.
11
,
333
418
(
1958
).
19.
P. D.
Lax
and
R. S.
Phillips
, “
Local boundary conditions for dissipative symmetric linear differential operators
,”
Commun. Pure Appl. Math.
13
,
427
455
(
1960
).
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