A critique of the modified Levermore–Pomraning equations

Available online at www.sciencedirect.com

www.elsevier.com/locate/anucene

Annals of Nuclear Energy 35 (2008) 446–457

annals of

NUCLEAR ENERGY

A critique of the modified Levermore–Pomraning equations

Richard Sanchez

Commissariat a l’Energie Atomique, DEN/DANS/DM2S, Service d’Etudes de Reacteurs et de Mathematiques Appliquees,

CEA de Saclay 91191, Gif-sur-Yvette cedex, France

Received 18 May 2007; accepted 5 July 2007Available online 24 October 2007

Abstract

The Modified Levermore–Pomraning (MLP) equations have been recently proposed by Akcasu as a treatment of transport in sto-chastic media with scattering. Using the bimaterial Markovian density statistics originally introduced by Pomraning, a new derivationof the MLP equations is presented for the continuous transport equation with energy dependence and ‘density’ sources. A heuristic MLPmodel is also introduced for calculations in finite slabs and is compared to the Levermore–Pomraning model using theoretical consid-erations. Numerical comparisons of the MLP and LP models are presented for a set of standard tests as well as related problems forwhich we give reference results.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

Transport in stochastic materials has applications inradiative transfer as well as reactor analysis. The relevantquantities of interest are the so-called material fluxes.These are the ensemble averaged fluxes over the set ofphysical realizations that have a given material in the samegeometrical location. A system of exact transport-like cou-pled equations for the material fluxes was independentlyderived by Adams et al. (1989) and by Sanchez (1989),but these equations are not closed. Correlation termsappear that depend on new ensemble averaged fluxes,called interface fluxes. These new fluxes are the averagesover the realizations that have one material to the left ofa geometrical location and a different material to the rightof the location, along a given direction of particle motion.When the material chord lengths follow Markov statisticsand the materials are purely absorbing one finds that thetransition flux equals the material flux for the upstreammaterial. This natural closure has been invoked to derivea set of closed equations for the general case of transportwith collisions. These equations are known as the Lever-more–Pomraning (LP) model (Levermore et al., 1988;

0306-4549/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.anucene.2007.07.017

E-mail address: [email protected]

Pomraning, 1991) and can be applied to a multidimen-sional geometry. For the purpose of this paper, we needonly the one-dimensional slab form for a bimaterial sto-chastic mixture:

ðlox þ AkÞhwik ¼ Sk þ jlj pk0pkkk0ðhwik0 � hwikÞ; L < x < R;

hwðx; vÞik ¼ winL;kðvÞ; x ¼ L; l > 0;

hwðx; vÞik ¼ winR;kðvÞ; x ¼ R; l < 0:

8>><>>:Here k = 1, 2 denotes one of the materials, k 0 = 3 � k

and Ak = Rk � Hk. The Rk(x), Hk(x) and Sk(x, v) are thetotal cross section, the scattering operator and the sourcefor material k, kk and pk are the mean chord length andthe probability for material k at position x, respectively,and, with v = (l, E), the Æw(x, v)æk are the material fluxes,obtained by ensemble averaging of the flux over the setof realizations Xk(x) that have material k at location x.

Recently, Akcasu has proposed a modification of the LPmodel. The modified MLP equations (Akcasu, 2006) havethe advantage of given exact results for a special half spaceproblem as well as for related full space problems. TheMLP equations have also been proved to give the atomicmix diffusion equation at the diffusive limit, even for finitechord lengths (Larsen, 2006). Although the model has beenanalyzed for use in finite rod problems, at present time

mailto:[email protected]

R. Sanchez / Annals of Nuclear Energy 35 (2008) 446–457 447

there is not a clear consensus as to how apply the MLPequations for more realistic finite-slab problems. One ofthe aims of this work is to analyze the straightforward

application of the MLP equations to finite slab problemsand to present a numerical comparison between LP andMLP for a set of standard problems (Zuchuat et al.,1994) as well as for related source problems. We stress thatin our approach we use the MLP equations as a heuristicmodel for a finite slab with the natural boundary condi-tions of the problem.

This paper is a revised and extended version of a worksubmitted to the recent M&C Topical Meeting in Monte-rey (Sanchez, 2007a). Akcasu (2007a) has also introducedan explicit integral equation for the ensemble average fluxin a slab based on a stochastic propagator and analyzedthe solutions for the rod problem in the case of no absorp-tion. This Stochastic Transition Matrix formalism is ana-lyzed in a companion paper (Sanchez, 2007b).

In the next section, we follow the early work by Pom-raning (Pomraning, 1991) to define the special class ofhalf space problems for which one can obtain exact resultsfor the ensemble average flux for bimaterial Markovianstatistics. In the following section, we use the formalismestablished by Pomraning, based on the probabilityPk(s;x)ds for a realization to have s (x, x) 2 (s, s + ds)conditioned to x 2 Xk(x), to give a new, shorter deriva-tion of Akcasu’s MLP equations. In Section 4, we givea comparative analysis of the LP and MLP models basedon already established and new theoretical results.Numerical results for the MLP heuristic model are pre-sented in Section 5. Finally, conclusions are given in thelast section. Related material has been relegated to theappendices. A short derivation of the Liouville masterequation for the Pk(s; x) is given in Appendix A. InAppendix B, we discuss a stable numerical algorithm forthe solution of the MLP model.

2. A restricted exact treatment

We consider linear particle transport in a half-space slabgeometry comprising a random material. We denote byX = {x, p(x)} the space of all physical realizations withp(x) P 0 and

RX pðxÞdx ¼ 1. Every realization x 2 X con-

sists of a triplet of functions x! (Rx, Rs,x, Sx), where,with v = (l, E), Rx(x, v) and Rs,x(x, v 0 ! v) are the totaland differential scattering cross sections and Sx(x, v) isthe external source. Therefore, the flux wx(x, v) satisfiesthe equation:

ðlox þ AxÞwx ¼ Sx; L < x <1;wxðx; vÞ ¼ win

L ðvÞ; x ¼ L; l > 0;

wxðx; vÞ <1; x!1;

8><>: ð1Þ

where Ax = Rx � Hx and Hx is the scattering operator

ðHxwxÞðx; vÞ ¼Z

dv0Rs;xðx; v0 ! vÞwxðx; v0Þ:

In order to establish exact results in the presence of scat-tering, Pomraning introduced a ‘restricted exact treatment’(Pomraning, 1991) by considering a homogeneous materialwith stochastic density such that the material properties(Rx, Rs,x, Sx) have the same spatial dependence:

Rxðx; vÞ ¼ NxðxÞrðvÞ;Rs;xðx; v0 ! vÞ ¼ NxðxÞrsðv0 ! vÞ;

ð2Þ

and

Sxðx; vÞ ¼ NxðxÞsðvÞ; ð3Þ

where Nx(x) is the stochastic density for the realization x,that now consists of a single function x! (Nx), withr(v) > 0 and rs(v

0 ! v), s(v) P 0 common to all realizationsin X. The two conditions in (2) give

Ax ¼ NxðxÞA; ð4Þ

where A ¼ r� h is now a deterministic operator and h is ascattering operator like H but with kernel rs(v

0 ! v). Weshall call these statistics density statistics and use the termdensity statistics and sources when condition (3) is also sat-isfied. In the following, we follow Pomraning and assumedensity statistics and sources.

By introducing the density (‘optical’) thickness

sðx;xÞ ¼Z x

L

NxðyÞdy ð5Þ

and, by changing variables x! s(x, x) so thatox! (oxs)os = Nx(x)os, one can write (1) in terms of theflux wðs; vÞ ¼ wxðx; vÞ:

ðlos þAÞw ¼ s; 0 < s <1;wðs; vÞ ¼ win

L ðvÞ; s ¼ 0; l > 0;

wðs; vÞ <1; s!1:

8><>: ð6Þ

Note that the deterministic flux wðs; vÞ is the same for allrealizations. Following Pomraning (1991) we define thedensity of probability P(s; x), such that P(s; x)ds is theprobability for a realization x to have s(x, x) 2 (s,s + ds). Then, the ensemble average flux at position x

can be written as

hwðx; vÞi ¼Z

Xwxðx; vÞpðxÞdx ¼

Z 1

0

wðs; vÞP ðs; xÞds: ð7Þ

The above result applies to a half space (HS) comprisinga homogeneous material with density statistics and sourcesand with deterministic incoming angular flux. We shallrefer to this problem as the HSSD problem. As an asidecomment we observe that P(s; L) = d(s) and that, there-fore, the ensemble average of the exiting angular flux forthe HSSD problem, hwðL; vÞi ¼ wð0; vÞ, is independent ofthe statistical set (Pomraning, 1991). A deeper result isobtained from wxðL; vÞ ¼ wð0; vÞ which shows that theboundary fluxes are deterministic. Notice that this propertyis necessary to prove that (wx,x) is a joint Markovian

448 R. Sanchez / Annals of Nuclear Energy 35 (2008) 446–457

process, a necessary condition for the derivation of theMLP equations from the jump probabilities associated tothis process (Van Kampen, 1992).

Pomraning considered a statistical set of binary homo-geneous Markov realizations with material probabilitiespk (k = 1, 2) and introduced the conditional probabilitiesfk(s; x)ds = probability for s(x, x) 2 (s, s + ds) given thatx lies in material k. Pomraning also derived analyticalexpressions for the fk(s;x). Hence, by replacingP ðs; xÞ ¼

P2k¼1pkfkðs; xÞ in Eq. (7) and by using the analyt-

ical expressions for the fk(s; x) one obtains an explicit for-mula to compute the ensemble average flux. Pomraningused this approach to analyze a number of half and fullspace problems (Pomraning, 1991).

In this work, we depart from Pomraning’s approach andfollow Akcasu (2006) to derive an implicit formula for theensemble average flux in the form of transport-like equa-tions for the material fluxes. For bimaterial statistics wewrite P ðs; xÞ ¼

P2k¼1P kðs; xÞ, where Pk(s; x)ds is the proba-

bility for s(x, x) 2 (s, s + ds) and x(x) = k. We have then

hwðx; vÞi ¼X2

k¼1

wkðx; vÞ;

where, with Xk(x) = {x 2 X, x(x) = k} denoting the set ofrealizations that have material k at x,

wkðx; vÞ ¼Z

XkðxÞwxðx; vÞpðxÞdx ¼

Z 1

0

wðs; vÞP kðs; xÞds:

ð8Þ

The basic idea to derive transport-like equations for theÆwæk is to differentiate (8) with respect to x to obtain

ðoxwkÞðx; vÞ ¼Z 1

0

wðs; vÞðoxP kÞðs; xÞds ð9Þ

and to introduce a dynamic equation for ox Pk. As we showin the next section, such a dynamic equation can be writtendown when the bimaterial statistics are Markovian. An-other possibility would be to compute the ox Pk from theanalytical expressions obtained by Pomraning for thefk(s; x) for the case of stationary bimaterial Markovstatistics.

Fig. 1. A realization for bimaterial Markov density statistics. The densitychanges from a material density to the other at positions where therealization has different material to the left and to the right.

3. Derivation of the MLP equations

In this section, we derive Akcasu’s MLP equation forthe continuous transport equation with energy dependenceand in the presence of density sources. Our derivation isbased in the transition probabilities Pk(s; x) associated tothe joint Markov process (s, x). In contrast, Akcasu’s der-ivation was done for the one-group, discrete ordinates formof the transport equation and was based on the joint Mar-kov process ðW!;xÞ, where the components of vector W

!ðxÞare the angular fluxes for all the discretized angular direc-tions (Akcasu, 2006, 2007a). A different derivation, basedon the expansion of the flux over Case’s eigenfunctions,was done recently by Larsen and Prinja (2007). We note

that the crucial future of all three derivations is the exis-tence of a joint Markov process whose associated transi-tion probabilities are then used to derive the MLPequations.

We consider a statistical set of realizations with bimate-rial Markovian chord distributions. Each material is char-acterized by a bounded deterministic density Nk(x) (k = 1,2), as in (2) and (3), and each realization x is a binary Mar-kov process with x(x) 2 {1, 2} that assigns materialk = x(x) to position x. Therefore, Nx(x) = Nk(x) forx 2 Xk (x). Fig. 1 shows an example of a realization.

We consider the general case of inhomogeneous statis-tics and denote by 1/kk(x) the transition probability perunit length from material k into material k 0 = 3 � k. Weobserve that the density distance (5) obeys the stochasticdifferential equation

ðoxsÞðx;xÞ ¼ NxðxÞ; L < x <1;with a deterministic initial condition s(L, x) = 0. Thereforethe pair (s, x) defines a Markov process and this allows toobtain a dynamic equation for the oxPk. Here we use thegeneral formulation established by Van Kampen (1992)to directly write down the Liouville master equation forthe Pk,

oxP k ¼ �osðNkP kÞ þ P k0=kk0 � P k=kk; s > 0;

P kðs; xÞ ¼ pkðxÞdðsÞ; x ¼ 0;

�ð10Þ

while in Appendix A we give a straightforward derivationof this formula. Next, we replace (10) in (9) to obtain

ðlox þ AkÞwk ¼ pkSk þ lðwk0=kk0 �wk=kkÞ; L < x <1;wkðx; vÞ ¼ pkðLÞwin

L ðvÞ; x ¼ L; l > 0;

wkðx; vÞ <1; x!1:

8><>:ð11Þ

where Ak ¼ N kA.To derive the MLP equations in (11) we have carried out

an integration by parts for the integralR1

0wðs; vÞ�

osðN kP kÞds and used Eq. (6) in the result. Note that the


boundary term wosðNkP kÞj10 cancels out because w isbounded and Pk(0; x) = Pk(1; x) = 0 for x 2 (L,1).

Finally, in the boundary conditions of (11),

pkðxÞ ¼Z

XkðxÞpðxÞdx ¼

Z 1

0

P kðs; xÞds

is the probability for material k at x. The dynamics for thepk are straightforwardly obtained by integrating (10) overs:

oxpk ¼ pk0=kk0 � pk=kk ð12Þwith given initial values pk(L) satisfying pkðLÞ þ pk0 (L) = 1.Note that the values pk(L) uniquely define the set of Mar-kovian realizations X.

4. Analysis of the MLP model

In this section, we compare the MLP model with naturalboundary conditions and the LP model, while in the nextsection we present a numerical comparison for somefinite-slab problems. The MLP model we want to analyzeis to solve the MLP equations with the natural boundaryconditions.

We write the MLP model equations for the ensembleaverage material fluxes Æw(x, v)æk = wk(x,v)/pk(x):

ðlox þ AkÞhwik ¼ Sk þ lpk0

pkkk0ðhwik0 � hwikÞ; L < x < R;

ð13Þwhere the last term in the right-hand-side contains the cor-relation contribution, with the boundary conditions

hwðx; vÞik ¼ winL;kðvÞ; x ¼ L; l > 0;

hwðx; vÞik ¼ winR;kðvÞ; x ¼ R; l < 0;

(ð14Þ

where winL;k and win

R;k are the ensemble average boundaryfluxes for material k.

As an aside, we mention that it is possible to define otherMLP models for finite slabs, using for example the righthalf-space form of the MLP equations for particles enter-ing the left side and the left half-space form for particlesentering the right side (Larsen, 2007). By using both formsof the MLP equations, such model can re-establish thesymmetry of the solution for a symmetric slab problembut, any one of the two component solutions used for thismodel is essentially equivalent to the simple heuristic modelwe propose here and, therefore, will suffer from the inher-ent numerical difficulties associated to MLP that we discussin this section. More generally, this will also apply to anyfinite-slab model that retains the basic characteristic ofthe MLP equations, that is, the sign of l in the correlationterm of Eq. (13).

4.1. The purely absorbing limit

The first point we shall discuss is the limiting case whenthere is no scattering. As it is well known, the LP model is

exact in this limit for Markovian chord statistics. Hence,one would expect that the LP solution for a small scatter-ing perturbation around this limit case will give accurateresults, while probably this will not be the case for theMLP model, which does not have the correct form at thezero-scattering limit. However, this raises the question ofwhy the MLP equations are correct for the HSSD problemwith no scattering. The answer, as we will show, is that inthis case the ensemble average material fluxes for negativel’s are constant and, therefore, independent of k, and thismakes the fluxes in the correlation term in Eq. (13) to can-cel one each other. For the HSSD problem we substituteboundary conditions (13) by

hwðx; vÞik ¼ winL ðvÞ; x ¼ L; l > 0;

hwðx; vÞik <1; x!1:

(For the purely absorbing case and for l < 0 the exact

correlation term, as predicted by LP theory, has a plus sign.Thus, for the MLP equations to be right one must havehwik0 ¼ hwik for l < 0.

Indeed, consider these equations for the HSSD problemwithout scattering. Note that the solution of Eq. (6) for noscattering and l < 0 is

wðs; vÞ ¼Z 1

0

e�rðvÞzsðvÞdz ¼ sðvÞ=rðvÞ; l < 0;

that, in view of (8), results in Æw(x, v)æk = s(v)/r(v), l < 0. Itis easily verified that indeed this is the solution of (13) forl < 0.

Another way to see this result is to note that for neg-ative l the fluxes at position x can be obtained from a‘modified’ HSSD problem for the half space with leftboundary at x > L that we describe with the spatial var-iable y, 0 < y <1. This new HSSD problem has asboundary values for the pk(y = 0) the pk(x) valuesobtained from Eq. (12). Next, we use the fact that theexiting boundary fluxes for the HSSD problem are deter-ministic so that for the modified HSSD problem, withs = 0 at position y = 0, we have Pk (s; 0) = pk(x)d(s)and we can write

wkðx; vÞ ¼ pkðxÞwð0; vÞ ! hwðx; vÞik ¼ wð0; vÞ; l < 0:

We conclude that for the purely absorbing HSSD prob-lem the MLP equations are correct because the correlationterms vanish for l < 0. A property that results from theparticular combination of random density bimaterial statis-tics and a half-space geometry (for which the exiting direc-tions have infinite-length trajectories).

However, for a purely absorbing finite slab the MLPheuristic model is exact only for particles propagatingtowards the right. For particles traveling to the left one has

wxðx; vÞ ¼ ½1� e�sxðx;RÞrðvÞ=jlj�sðvÞ=rðvÞ; l < 0;

where sxðx;RÞ ¼R R

x NxðyÞdy, and, therefore the left exitingflux is not deterministic.


4.2. Uniform solutions

An instinctive reaction when faced with a differentialequation is to look for uniform solutions. Both the LPand the MLP models accept uniform solutions of the formÆw(x, v)æk = w(v) under the conditions that (AkÆwæk)(x,v) = Sk(x, v) and that the entering boundary fluxes areequal to w(v). However, for these solutions to be meaning-ful, they should correspond to exact material averages.Hence, the flux for each realization should also be equalto w(v) which, in turn, implies (Axwx)(x, v) = Sx(x,v) andwin

L;xðvÞ ¼ winL;xðvÞ ¼ wðvÞ. Clearly, these last conditions are

satisfied for density statistics and sources and we concludethat LP and MLP give the exact uniform solution w(v) fordensity statistics (not necessarily Markovian) for finiteslabs problems with incoming fluxes equal to w(v) and withdensity sources, as in (3), such that

sðvÞ ¼ ðAwÞðvÞ

Of course, this property holds for LP in general geome-try for which density statistics and sources are defined witha density Nx(r). As an aside comment we should mentionthat for years we have been using this type of uniform solu-tions of the transport equation as a means to effectuatesimple checks of transport codes.

4.3. The diffusion limit

An extensive analysis of the behavior of the Levermore–Pomraning (LP) model in the diffusion limit was presentedin Malvagi et al. (1992). The authors took stock of the dif-fusion limit of the transport equation with the scaling

R � Oð1Þ; lox � Oð�Þ; Ra; S � Oð�2Þand considered a variable scaling for the Markovian tran-sition probabilities kk � O(�n). They found four differentbehaviors: weak coupling (n < �1), moderate coupling(n = �1), strong coupling (n = 0) and atomic mix (n > 0).Their results show that both, strong coupling and atomicmix, yield an asymptotic diffusion equation for the homo-geneous mixture. However, while atomic mix predicts theexpected diffusion coefficient, D = 1/(3 < R > ), for strongcoupling one finds a different diffusion coefficient, whichdepends on the Rk and the kk values (Malvagi et al., 1992).

Recently, Larsen et al. (Larsen et al., 2005) revisited thediffusion limit for the LP equations for the case of strongcoupling and atomic mix (n P 0) and found the sameresults as in the previous study by Malvagi et al. However,Larsen et al. analyzed also the diffusion limit of the trans-port equation for a deterministic medium composed ofalternate lamps of materials with widths lk. For strong cou-pling, lkRk � O(1), they found that the transport equationlimits to the diffusion equation with the familiar diffusioncoefficient D ¼ 1=ð3RÞ, where R is the total cross sectionof the homogenized material:

�ox1

3RoxUþ RaUþ Q ¼ 0: ð15Þ

Because this result applies to most of the realizationsone concludes that the ensemble average flux obeys (15)with the ensemble averaged cross sections and sources.The analysis was confirmed by numerical experiments thatshowed that for strong coupling, kkRk � O(1), binary Mar-kovian statistics limits to the classical diffusion equation,while LP and its diffusion limit yield a different result (Lar-sen et al., 2005). Lately, Larsen analyzed the MLP equa-tions (Larsen, 2006) in the diffusion limit with strongcoupling and found that MLP limits to the diffusion equa-tion with the ensemble averaged diffusion coefficient D = 1/(3ÆRæ) predicted by atomic mix limit; this contrasts with theresult previously obtained for the LP model (Malvagi et al.,1992; Larsen et al., 2005). The conclusion is that the MLPequations are also exact (while LP is not) at the diffusionlimit with strong coupling (Larsen, 2006).

4.4. Right boundary condition

Because of the unusual sign of the correlation term fornegative l values, there is a doubt as to whether theMLP equations should be only used with zero right incom-ing flux for finite slabs. To see that this should not alwaysbe the case, consider the two following problems:

(A) The HSSD problem in (L, 1).(B) A finite slab problem obtained by restricting every

realization of problem A to (L, R) and by addingthe appropriate right boundary condition: (wx)B

(R, v) = (wx)A(R, v) for l < 0.

Clearly, these two problems have the same materialfluxes Æw (x, v)æk in (L, R). Moreover, the MLP modelfor problem B gives the exact values for the (L, R) if oneuses as right boundary condition

hwBðR; vÞik ¼ hwAðR; vÞik; l < 0; ð16Þwhere ÆwA(R, v)æk are the exact material fluxes (as given, byexample, by the MLP equations for problem A). Noticethat this boundary condition depends on the materialand, therefore, is not deterministic.

We conclude that for this finite-slab problem with den-sity statistics and sources the MLP model is exact if oneuses as right boundary condition the non-zero fluxes in(16). This property has been numerically verified for thelimit case with no scattering for which the boundary fluxes(wx)A(R, v) = s(v)/r(v) for l < 0 are deterministic.

Although somewhat artificial, this example shows thatthe MLP equations can also be exact for a particularfinite-slab problem. This is only possible when the problemis a ‘restriction’ of the half-space problem for which MLPis exact. This example sustains the opinion of one of thereviewers who said ‘the MLP equations are irrevocably

linked to the halfspace’. A point of view that we fully share


with the reviewer and that, to our believe, is at the root ofthe difficulties of finding an MLP model that will work forrealistic finite-slab problems.

4.5. Symmetry

Consider a finite slab for x 2 (�a, a) with symmetricboundary conditions, win(�a, v) = win(a, �v) for l > 0,comprising a homogeneous, symmetrical bimaterial Mar-kovian mixture with constant material properties. Then,for any realization x with density p(x) there is a symmet-rical realization x 0 = sx such that p(x 0) = p(x) andx 0(�x) = x(x) and, by the symmetry of the boundary con-ditions, wx0 ð�x;�vÞ ¼ wxðx; vÞ. Clearly, Xk(�x) = sXk(x)and we conclude

hwðx; vÞik ¼Z

XkðxÞwxðx; vÞpðxÞdx

¼Z

Xkð�xÞwxð�x;�vÞpðxÞdx ¼ hwð�x;�vÞik:

This symmetry is preserved by the LP equations but failsfor the MLP model.

4.6. The MLP equations for a left half-space

The lack of symmetry is also apparent when one writesthe MLP equations for a left half-space (Akcasu, 2006;Larsen and Prinja, 2007). For completeness, we review thispoint here. Consider a set of realizations X = {x, p(x)}over the spatial domain (�1, R). The transport equationfor a realization is like that in Eq. (1) except that nowx 2 (�1, R) and the flux wx(x, v) obeys the right boundarycondition wxðR; vÞ ¼ win

RðvÞ for l < 0 and remains boundedat infinity, limx!�1 wx(x, v) <1. Instead of repeating ourderivation for this case, we introduce the (symmetric)change of variables (x, v)! (L + R � x, �v) so that theflux ewxðx; vÞ ¼ wxðLþ R� x;�vÞ obeys Eq. (1) with

winL ðvÞ ¼ win

Rð�vÞ, eSxðx; vÞ ¼ SxðLþ R� x;�vÞ and eAx ¼eRx � eH x, where eRxðx; vÞ ¼ RxðLþ R� x;�vÞ, andeRs;xðx; v0 ! vÞ ¼ Rs;xðLþ R� x;�v0 ! �vÞ. For densitystatistics and sources every realization is now a mapex : ½L;1� ! f1; 2g with exðxÞ ¼ xðLþ R� xÞ, and we haveeN xðxÞ ¼ NxðLþ R� xÞ, erðvÞ ¼ rð�vÞ, ersðv0 ! vÞ ¼rsð�v0 ! �vÞ and esðvÞ ¼ sð�vÞ. Then, for bimaterial Mar-kov statistics we obtain MLP Eq. (11) for the materialfluxes ewkðx; vÞ. Finally, by changing back to the originalvariables, (x, v)! (L + R � x, �v) we obtain the MLPequations for a left half-space:

ðlox þ AkÞwk ¼ pkSk � lðwk0=kk0 � wk=kkÞ; �1 < x < R;

wkðx; vÞ ¼ pkðRÞwinRðvÞ; x ¼ R; l < 0;

wkðx; vÞ <1; x! �1:

8><>:ð17Þ

In these equations pk(x) is the probability for material k atx 2 (�1, R) and dx/kk(x) is the jump probability frommaterial k at x into material k 0 in the interval (x � dx, x)

(Akcasu, 2006). Also, because of the final change of vari-ables x! L + R � x, the dynamic equation for the pk is�oxpk ¼ pk0=kk0 � pk=kk, �1 < x < R, with the ‘initial’ con-dition pk(R). We note that the MLP equations for a lefthalf-space are similar to those for a right half-space exceptfor the negative sign of l in the correlation term. As thechange of variables used to obtain the equations shows,the sign change l!�l is due to the lack of symmetryof the MLP equations.

The last problem for which the MLP equations are exactis that of a full space with material Markov density statis-tics and sources (Pomraning, 1991) and a deterministicdelta source at the origin x = 0:

ðlox þ AxÞwx ¼ Sx þ f ðvÞdðxÞ; �1 < x <1;wxðx; vÞ <1; jxj ! 1:

�By measuring distance in terms of the optical density

sðx;xÞ ¼R x

0 NxðyÞdy one has

ðlos þAÞw ¼ sþ f ðvÞdðsÞ; �1 < s <1;wðs; vÞ <1; jsj ! 1:

(This shows that the flux wðs; vÞ is deterministic, which im-plies that wxð0; vÞ ¼ wð0; vÞ is also deterministic. We con-sider now this problem as two coupled half-spaceproblems, for x < 0 and for x > 0. Because the flux enteringeach half space is deterministic, one concludes that theMLP equations for the left and right half-spaces, Eqs.(17) and (11), predict the exact ensemble and material aver-age fluxes for the problem. This conclusion was arrived toby Pomraning (1991) and Akcasu (2006), who consideredhomogeneous symmetric statistics for a symmetric one-group rod problem with a symmetric delta source at theorigin and no density sources.

4.7. Positiveness

Because the collision operator is positive, the analysis ofpositiveness may be done considering a punctual source ora boundary flux (a surface source). We consider the follow-ing LP or MLP equation with constant cross sections anduniform statistics:

ðlox þ BÞw!¼ dðx� x0Þ S!; L 6 x; x0 6 R; ð18Þ

where w!¼ fhwik; k ¼ 1; 2g represents the material fluxes

for a bimaterial Markovian mixture, S!ðlÞ is the source

and

B ¼b1 �c2

�c1 b2

� �: ð19Þ

Here, for the MLP model, bk = Rk + l/kk and ck = l/kk,while for the LP model one has to change l by jlj in theseexpressions.

For simplicity we assume that the cross sections are con-stant and the statistics uniform so that B is a constantmatrix. Then, the solution of (18) is


w!ðx; lÞ ¼ 1

2sgðx� x0Þe�Bðx�x0Þ=l S

!ðlÞl

;

where sg(x) is the sign of x.Because (x � x0)/l, sg(x � x0)/l > 0, positiveness rest

on the matrix T ¼ e�Bðx�x0Þ=l. By using the fact that B canbe diagonalized (see Appendix B) we can write:

T ¼ MBtþ

t�

� �M�1

B ;

where t� ¼ e�z�ðx�x0Þ=l, z+ > z� are the eigenvalues of B and

MB ¼c2 c2

b1 � zþ zþ � b2

� �: ð20Þ

Here MB is the matrix, w!¼ MB n

!, that gives the compo-

nents of a vector on the canonical basis, ek, in terms ofits components n

!¼ fnk; k ¼ 1; 2g on the basis of the eigen-vectors of matrix B, e± = c2e1 + (b1 � z±)e2,

After a little bit of algebra we obtain

T ¼ 1

D

ðtþ þ t�Þzþ � ðtþb2 þ t�b1Þ ðt� � tþÞc2

ðt� � tþÞc1 ðtþ þ t�Þzþ � ðtþb1 þ t�b2Þ

� �;

where D2 is the discriminant in (23).By using the fact that z+ > max(b1, b2) we see that the

diagonal terms of T are positive. Also, t� > t+ and forthe LP model, for which ck > 0, the off-diagonal termsare also positive. On the other hand, for the MLP modelthe off-diagonal elements of T are positive for l > 0 andnegative for l < 0. This proves that, with positive sourcesand boundary fluxes, the LP model will always give posi-tive fluxes, while the MLP model adds negative compo-nents to the fluxes, for the sources, boundary fluxes orcollisions at positions x0 > x. Thus, the MLP model isnot unconditionally positive.

4.8. Summary of comparisons

The previous comparisons can be summarized asfollows:

� The LP model is inherently positive and can be appliedto any type of geometry, not necessarily slab, it is exactwith no scattering, satisfies the atomic mix limit and hasthe correct form for the diffusion asymptotic limit foratomic mix, while it does not preserve the diffusion limitfor strong coupling. On the other hand, numericalexploration has shown that LP is a robust model withpredicts the correct ensemble average fluxes with errorsof the order of 5–10% (Levermore et al., 1988; Zuchuatet al., 1994).� The MLP model is exact with scattering for the HSSD

problem and related left half-space and full space prob-lems. Otherwise it is never exact with no scattering forparticles with l < 0 (except for artificial problems‘extracted’ from the HSSD problem). This model pre-dicts the atomic mix diffusion limit for both strong

and atomic mix couplings and preserves the infinite-medium relaxation modes (however, for a detailed dis-cussion of this point see (Sanchez, 2007b)). The modelpresents an asymmetry in the treatment of the fluxes indirections l > 0 and l < 0 that prevents it to preservethe symmetry of the solution when the boundary condi-tions and statistics are symmetric. Also, the model is notinherently positive and may produce negative fluxes.

5. Numerical examples for the MLP model

In Appendix B, we have analyzed the numerical solutionof the discrete ordinate formulation of the MLP heuristicmodel in a finite slab with uniform cross sections and sta-tistics. The stability of the solution depends on the natureof the eigenvalues z± of matrix B in Eq. (19). Whilez+ > 0, for the smallest eigenvalue we find z� < 0 wheneversmax 6 2 or smin 6 smax/(smax � 1), where sk = Rkkk is themean chord optical length for material k. Moreover, asshown in the appendix, a brutal inversion of the entireoperator does not always result in a convergent iterativescheme. Nevertheless, we have chosen to apply a straight-forward direct inversion with forward source iterationsand a simple diamond differencing, while counting on dou-ble precision arithmetics to attenuate the instability effectsand force convergence, when possible, by diminishing thewidth of the spatial mesh. As a consequence of the afore-mentioned problems some of the calculations we run didnot converge.

We have run a number of calculations for the problemsuite defined by Pomraning for albedo and transmissioncalculations with a unit isotropic flux illumination on theleft boundary of a finite slab (Zuchuat et al., 1994), as wellas related source problems. All the results were done with aS16 angular approximation and converged to 10�5 in rela-tive precision. The number of spatial cells was automati-cally adjusted to satisfy a conservative positive criterionso that the cell width satisfied Dcell 6 jlminj/(5Rmax). Thestability and convergence of the MLP model solutionswas verified by running the calculations with 20,000 regionsto obtain identical results. Reference results were obtainedfrom the averaging of 50,000 calculations for randomlygenerated realizations.

We consider first two albedo-transmission problemsfrom Pomraning’s suite for a slab of length L = 10. Thedata for problem 1 are R1 = 10/99, R2 = 100/11, c1 = 1,c2 = 0, k1 = 99/10 and k2 = 11/10, while for problem 2,R1 = 2/101, R2 = 200/101, c1 = 0.9, c2 = 0.9, k1 = 101/20and k2 = 101/20. We note that problem 2 is a problem withdensity statistics, the very class of statistics for which theMLP model is exact for a half-space. Results for theseproblems are shown in Table 1 and Figs. 2–7. Curiouslyenough, the results show that MLP is better than LP forproblem 1 while it is worse than LP for problem 2.

Finally, in Table 2 and Figs. 8–10, we present results fora slab problem (problem 3) with density sources. The data

Table 1Albedos b and transmissions t for problems 1 and 2

Problem Reference LP MLP

1 b 4.34546 · 10�1 2.90962 · 10�1 3.11812 · 10�1

t 1.85351 · 10�1 1.94468 · 10�1 1.94569 · 10�1

2 b 4.46270 · 10�1 3.27217 · 10�1 1.05145 · 10�1

t 1.04333 · 10�1 1.19473 · 10�1 1.33489 · 10�1

10

8

6

4

2

0

100806040200

REFLPMLP

Fig. 2. Problem 1. Ensemble average fluxes.

10

8

6

4

2

0100806040200

REFLPMLP

Fig. 3. Problem 1. Material fluxes for material 1 (w1).

10

8

6

4

2

0100806040200

REFLPMLP


10

8

6

4

2

0100806040200

REFLPMLP


10

8

6

4

2

0100806040200

REFLPMLP


10

8

6

4

2

0100806040200

REFLPMLP


Table 2Problem 3: Left and right exiting currents

Jout Reference LP MLP

Left 5.69116 · 10�1 7.00687 · 10�1 �3.11368Right 5.69116 · 10�1 7.00687 · 10�1 9.64264 · 10�1


is the same as for problem 2, except that the entering fluxesare zero and the sources are S1 = 0.01 and S2 = 1.

Although for problem 2 the MLP ensemble and materialfluxes are positive, we observed that 40% of the angularfluxes were negative at convergence. This value increasedto 52% for problem 3, for which the scalar fluxes becomenegative. Propagation of negative fluxes in MLP is due toparticles emitted in negative directions by the sources

(scattering of externals). For problem 2, particles enterthe left side of the slab and the main contribution to thescalar fluxes is from particles moving with positive direc-tions, so the scalar fluxes remain positive. However, forproblem 3 the particles appear uniformly and isotropicallyin the slab and therefore particles traveling left can over-come those traveling right and the result are negative scalarfluxes on the left of the slab, as shown in Figs. 8 and 9. Thisdifference in behavior can also be explained in terms of the

10

8

6

4

2

0100806040200

REFLPMLP


-8

-6

-4

-2

0

2

4

100806040200

REFLPMLP


-20

-15

-10

-5

0

5

100806040200

REFLPMLP



MLP eigenvalues (see Appendix B). In Fig. (11) we havereproduced the eigenvalues for problems 1 and 2. To everyl corresponds two eigenvalues, k� and k+, k� < k+. Forproblem 1 the k� branch have slightly negative values forthe last 3 negative l directions, however these values are

10

8

6

4

2

0

-1.0 -0.5 0.0 0.5 1.0

2.5

2.0

1.5

1.0

0.5

0.0

-0.5

Fig. 11. MLP eigenvalues versus l for a S16 angular discretization

largely dominated by those in the k+ branch. On the con-trary, for problems 2 and 3, the lower branch becomes neg-ative much earlier and reaches negative values that cannotbe neglected as compared to those in the highest branch.

6. Conclusions

Our numerical comparisons between the LP model andthe MLP heuristic model with natural boundary conditionsshow that MLP behaves well when the bulk of the particlesmove in positive directions, although we observed a fairamount of negative angular fluxes. However, in the pres-ence of uniform and isotropic sources, the lack of positive-ness and symmetry of MLP shows deleterious effects in theform of negative fluxes. Akcasu (2007a,b) proposes to usedifferent boundary conditions to compute source problems,but then the MLP solution does not exactly satisfy the nat-ural boundary conditions of the problem and, although theensemble averages of the entering currents are zero, thematerial averages of the entering currents on the right sur-face are non zero. Furthermore, the material fluxes canbecome negative (Sanchez, 2007b). It is clear that muchwork needs to be done in order to define approximatedboundary conditions that give satisfactory results whenusing MLP in finite slabs.

Our opinion is that it would be interesting to obtain animproved model from the LP and MLP equations. Such amodel should retain the robust properties of LP (generalgeometry, symmetry, positiveness, exact non scatteringlimit) while adding a better treatment for collisions andpreserving the MLP equations for the HSSD problem.

A way to achieve this could be to derive a better closureof the exact equations (Adams et al., 1989; Sanchez, 1989)by using MLP to obtain an improved approximation forthe interface fluxes hwðx; lÞik!k0 , where hwðx; lÞik!k0 is theensemble average of the flux over all realizations that tran-sit at x in direction l from material k into material k 0.

For transport in purely absorbing materials one has therelation hwðx; lÞik!k0 ¼ hwðx; lÞik, a closure that was usedto derive the LP equations from the exact ones. On theother hand, by comparing the exact equations with MLPfor the HSSD problem, one realizes that, in this particularsituation,

-1.0 -0.5 0.0 0.5 1.0

. On the left for problem 1, on the right for problems 2 and 3.


hwðx; lÞik!k0 ¼ hwðx; lÞik; l > 0;

hwðx; lÞik!k0 ¼ hwðx; lÞik0 ; l < 0:ð21Þ

This is an interesting finding in itself, but this property isnot satisfied by finite slabs of for half-slabs with other typeof statistics different from density statistics and sources.Formula (21) holds only for the HSSD problem, and it isnot clear how this property can be used to define a closurefor the interface fluxes hwðx; lÞik!k0 for more realisticproblems.

A simple approach may consist of constructing a heuris-tic model based on an interpolation formula of the follow-ing form:

hwðx; lÞik!k0 ¼ hwðx; lÞik; l > 0;

hwðx; lÞik!k0 ¼ akk0 hwðx; lÞik þ ð1� akk0 Þhwðx; lÞik0 ; l < 0;

where akk0 is a function of the material properties, meanchord lengths and width L of the slab such that0 6 akk0 6 1, akk0 ¼ 1 when ck ¼ ck0 ¼ 0 and akk0 ¼ 1 in theconditions of the HSSD problem. The simplest functionof this type is akk0 ¼ ae�LR=ðae�LR þ ck þ ck0 Þ, where a > 0is a parameter. Such an interpolation will give the goodlimit for pure absorption and for the HSSD problem.

Acknowledgements

Many thanks are due to Ed Larsen and Ziya Akcasu forcomments and discussions, as well as for providing unpub-lished material. Although there has been much controversyand our opinions are sometimes drastically different, theemail exchanges helped me to understand better MLP.Clearly, the opinions advanced in this paper are the soleresponsibility of the author.

Appendix A. Derivation of the Liouville master equation for

the Pk(s; x)

Consider the value of Pk(s; x + Dx). Because (s, x) is aMarkovian process, for Dx small, we can write Pk(s;x + Dx) in terms of Pk and P k0 at x. The realizations that con-tribute to Pk(s; x + Dx) are those that have material k atx + Dx with s(x, x) 2 (s, s + ds). These realizations eitherhave material k at x and keep this material in (x, x + Dx) withprobability 1 � Dx/kk, or have material k 0 at x and change tomaterial k in (x, x + Dx) with probability Dx/kk 0. Thus

P kðs; xþ DxÞ ¼ P kðs� NkDx; xÞð1� Dx=kkÞþ P k0 ðs� NkDx; xÞDx=kk0 þ OððDxÞ2Þ:

Next, we subtract Pk(s; x) from both sides of this expression,divide by Dx and take the limit for Dx! 0. The result is

ðoxP kÞðs; xÞ ¼ limDx!0þ

1

Dx½P kðs; xþ DxÞ � P kðs; xÞ�

¼ �P kðs; xÞ=kkðxÞ þ P k0 ðs; xÞ=kk0 ðxÞ

� limDx!0þ

1

Dx½P kðs; xÞ � P kðs� N kDx; xÞ�:

Finally, by recognizing that the limit on the right-hand-side can be written as limDs!0þ

1Ds N k½P kðs; xÞ � P kðs� Ds; xÞ�

we obtain the final expression:

ðoxP kÞðs; xÞ ¼ � P kðs; xÞ=kkðxÞþ P k0 ðs; xÞ=kk0 ðxÞ � osðNkP kÞðs; xÞ:

Appendix B. Numerical solution for the MLP model

We investigate the solution of the one-group MLPmodel equations by source iterations based on a discreteordinates approximation. By introducing the two-dimen-sional vectors w

!¼ fhwki; k ¼ 1; 2g and q!¼ fH khwkiþSk; k ¼ 1; 2g we write the power iteration process as

ðlox þ BÞw!n ¼ q!n�1; ð22Þwhere n is the iteration index and, for homogeneous statis-tics and cross sections, B is the matrix in (19).

B.1. Stability

Next, to investigate the stability of the iterations, welook for the eigenvalues of matrix B. These are given byz± = (b ± D)/2 where

b ¼ R1 þ R2 þ lð1=k1 þ 1=k2Þ;c ¼ R1R2 þ lðR2=k1 þ R1=k2Þand

D2 ¼ b2 � 4c

¼ ½R1 � R2 � lð1=k1 � 1=k2Þ�2 þ 4l2=ðk1k2ÞP 0: ð23Þ

Thus, the two eigenvalues are real and different and B isdiagonalizable.

We consider first the case of the LP model where one hasto replace l by jlj in the expressions for b, c and D. For thiscase one finds that the two eigenvalues are positive and,therefore, the stable power iteration strategy for (22) is tosolve for increasing x values for l > 0 and for decreasingx values for l < 0.

However, for the MLP operator we have two differentcases that depend on the signs of c and b. Let us definethe mean optical chord length sk = Rkkk. Then, coefficientsc and b are negative in the following ranges

c < 0; l < lc ¼ �s1s2

s1 þ s2

;

b < 0; l < lb ¼ �ðp2s1 þ p1s2Þ;where lb < lc < 0. We have, therefore, two differentdomains:

(a) For l > lc the two eigenvalues are positive and thestable strategy is to do a forward sweep.

(b) For l < lc one eigenvalue is positive and the othernegative. In this case, Eq. (22) has to be diagonalizedand the sweep done independently for each eigen-value. The stable strategy for the positive eigenvalueis the forward sweep, while the stable strategy for


the negative eigenvalue is the backward sweep. Atl = lc the smallest eigenvalue is 0, while at l = lb

the two eigenvalues are equal and of opposite sign.

For a negative eigenvalue, stability requires sweepingalong the backward trajectory starting with an exitingangular flux. Because this flux is not known one will haveto set up a shooting technique that, via iterations, will pro-duce the good incoming flux at the opposite boundary. Byusing linearity this requires only two iterations. In the first,one runs the backward sweep from left to right setting theinitial escaping flux wout

L to 1, and computes separately thecontributions from the internal sources and from wout

L ¼ 1to the flux win

R entering the opposite boundary: winRðqÞ and

winRðwout ¼ 1Þ. The correct exiting flux value is thus given

by woutL ¼ ðw

inR � win

RðqÞÞ=winRðwout ¼ 1Þ. It suffices then to

redo the backward sweep using the correct value of woutL .

Because this procedure for stable iterations is cumbersome,in a first time we shall use the standard iteration procedurethat sweeps always along the forward direction, startingwith the known boundary fluxes. For a negative eigenvaluek this gives for the transmitted flux:

wðx; lÞ ¼ ekðR�xÞ=lwinRðlÞ; l < 0;

where the positive exponential magnifies any errors due tofinite precision arithmetic operations. Note, however, thatthe relative precision of the solution,

dw=w ¼ dwinR=ðw

inR þ dwin

RÞ;remains bounded. Therefore, for finite slabs, for which themagnifying factor is bounded, it is possible, by increasingthe precision of the calculation if necessary, to obtain cor-rect results. If an increase of the precision (number of digitsused in the calculation) does not stabilize the problem, thenone can resort to the more complicated shooting schemefirst discussed.

B.2. Convergence of power iterations

Consider the discrete-ordinates diamond-differencingform of Eq. (22) with a mesh of constant width D:

2jljDþ B

� �w!

n ¼2jljD

w!

n;� þ q!n�1;

w!

n;þ ¼ 2 w!

n � w!

n;�;

ð24Þ

where w!

n and q!n�1 are cell-averaged values and w!

n;� arethe cell exiting ( + ) or entering (�) fluxes. In our imple-mentation of source iterations we have adopted the previ-ous scheme. Contrarily to the LP model, for l < 0 theMLP model weakens the diagonal terms of matrix 2jlj

D þ Band this may result in non convergence of the iterations.Several techniques can be used to resolve this problem:Wrapping around a Krylov solver (Fichtl et al., 2006),adopting a convergent splitting B = B+ � B� and iterating

only on term B�w!

n�1 and, more simply but less efficient,decrease the cell width D. To use a Krylov solver (GMRES)one would write the iterations as U

!¼ HL�1ðU!þ S!Þ,

where L = 2jlj/D + B, H is the scattering operator, S!

thesource and U

!¼ H w!

are the angular flux moments. How-ever, in our tests we have opted for the expedient solutionof diminishing the cell width.

B.3. Fourier analysis

This is another way to look at convergence. For an infi-nite slab of identical, homogeneous cells we introduce the

Fourier ansatzs w!

nðx; lÞ � w!

nðx; lÞ � expðixDnÞ and

w!

n;þðx; lÞ � w!

n;bdðX; lÞ expðiXDnÞ for the cell averagedflux and the exiting fluxes for cell n to obtain:

U!

nðzÞ ¼ MðzÞcU!

n�1ðzÞ;where we have assumed isotropic scattering q!ðx; lÞ ¼cRU!ðxÞ with c = diag{c1, c2}, R = diag{R1, R2}, U

!ðxÞ ¼ð1=2Þ

R 1

�1 w!ðx; lÞdl, z = (xD)/2 and MðzÞ ¼ 1

2

R 1

�1 Mðz; lÞdl. In the last expression the matrix M(z, l) is

Mðz; lÞ ¼ 1

Dðz; lÞ1þ lða2 þ ib2Þ la1

la2 1þ lða1 þ ib1Þ

� �with ak = 1/sk, bk = f(z)/(RkD), f(z) = 2tanz and

Dðz; lÞ ¼ 1þ lða1 þ ib1 þ a2 þ ib2Þ þ l2½iða1b2 þ a2b1Þ � b1b2�:

The function f(z) is periodic with period p and has thesymmetry property f(z) = �f(p � z). The function behavesas z for z! 0 and increases monotonously from 0 at z = 0to1 at z = p/2 where it has a vertical asymptote. Becauseof the periodicity the spectral radius of the iterator is givenby qM = max06z6p(j k+(z)j,jk�(z)j). In the last expressionk±(z) = (b ± K)/2 are the eigenvalues of M(z)c with

b ¼ ðc1 þ c2ÞI0 þ ½c1ða2 þ ib2Þ þ c2ða1 þ ib1Þ�I1;

K2 ¼ ½ðc1 � c2ÞI0 þ ½c1ða2 þ ib2Þ � c2ða1 þ ib1Þ�I1�2

þ 4c1c2a1a2I21;

where IpðzÞ ¼ ð1=2ÞR 1

�1 lpdl=Dðz; lÞ.From here on the analysis has to be done by numerical

evaluation and is out of the scope of this work. An analyt-ical result can be obtained for the flat modes (x = 0) whenc1 = c2 = c: For D > 0 let us look at the eigenvalues k±(z) atz = 0. One finds k+(0) = c and k�(0) = cI0(0) = (c/2)ln[(1 + x)/(1 � x)], where x = a1 + a2. Thus, with equalck values, power iterations will converge the flat modes inan infinite slab if 1/s1 + 1/s2 < (e2 � 1)/(e2 + 1) . 0.7616.Another interesting case, regarding Pomraning’s test prob-lems, is for c1 = 0. Here, for D > 0 and z = 0 we findk+(0) = c2 [s1I0(0) + s2]/(s1 + s2) and k�(0) = 0.

References

Adams, M.L., Larsen, E.W., Pomraning, G.C., 1989. Benchmark resultsfor particle transport in a binary markov statistical medium. J. Quant.Spectrosc. Radiat. Transfer 42, 253.

Akcasu, A.Z., 2006. A critique of the Levermore–Pomraning equations.Trans. Am. Nucl. Soc. 95, 545–546.

Akcasu, A.Z., 2007a. Modified-Levermore–Pomraning equation: itsderivation and its limitations. Ann. Nucl. Energy 34 (7), 579–590.


Akcasu, A.Z., 2007b. Personal communication.Fichtl, E.D., Warsa, J.S., Prinja, A.K., 2006. Krylov acceleration for

transport in binary statistical media. Trans. Am. Nucl. Soc. 95, 556–559.Larsen, E.W., 2006. Asymptotic diffusion limit of a modified Levermore–

Pomraning theory. Trans. Am. Nucl. Soc. 95, 547–549.Larsen, E.W., 2007a. Personal communication.Larsen, E.W., Prinja, A.K., 2007. A Derivation of Akcasu’s ‘MLP’

equations for 1-D particle transport in stochastic media. In: Proc.American Nuclear Society Topical Meeting: M&C 2007, Joint Inter-national Topical Meeting on Mathematics and Computation andSupercomputing in Reactor Applications, Monterey, CA,USA, Sep-tember 12–15.

Larsen, E.W., Vasques, R., Vilhena, N.T., 2005. Particle Transport in the1D Diffusive Atomic Mix Limit. In: Proc. American Nuclear SocietyTopical Meeting: M&C 2005, International Topical Meeting onMathematics and Computation, Supercomputing, Reactor Physicsand Nuclear and Biological Applications, Avignon, France, September12–15.

Levermore, C.D., Pomraning, G.C., Wong, J., 1988. Renewal theory fortransport processes in binary statistical mixtures. J. Math. Phys. 29, 995.

Malvagi, F., Pomraning, G.C., Sammartino, M., 1992. Asymptoticdiffusive limits of transport in Markovian mixtures. Nucl. Sci. Eng.112, 199–214.

Pomraning, G.C., 1991. Linear Kinetic Theory and Particle Transport inStochastic Mixtures. World Scientific, Singapore.

Sanchez, R., 1989. Linear kinetic theory in stochastic media. J. Math.Phys. 30, 2511–2948.

Sanchez, R., 2007a. A critique of the Modified Levermore–Pomraningmethod. In: Proc. American Nuclear Society Topical Meeting: M&C2007, Joint International Topical Meeting on Mathematics andComputation and Supercomputing in Reactor Applications, Monte-rey, CA,USA, September 12–15.

Sanchez, R., 2007b. A critique of the stochastic transition matrixformalism. Ann. Nucl. Energy 35, 458–471.

Van Kampen, N.G., 1992. Stochastic Processes in Physics and Chemistry.North Holland, Amsterdam.

Zuchuat, O., Sanchez, R., Zmijarevic, I., Malvagi, F., 1994.Transport in renewal statistical media: benchmarking and com-parison with models. J. Quant. Spectrosc. Radiat. Transfer 51,689–722.

A critique of the modified Levermore–Pomraning equations

Documents

Transcript of A critique of the modified Levermore–Pomraning equations