R. Sentis- A Short Survey of Numerical Methods for Radiative Transfer Problems

8/3/2019 R. Sentis- A Short Survey of Numerical Methods for Radiative Transfer Problems

1/16

JOURNAL DE PHYSIQUEColloque C7, suppldment au n012, Tome 49, ddcembre 1988

A SHORT SURVEY OF NUMERICAL METHODS FOR RADIATIVE TRANSFER PROBLEMS

R. SENTISC.E.A. Limeil, BP 27, F-94290 Villeneuve Saint Georges, FranceResume - On passe en revue les methodes numeriques couramment utilisees dans les probl6mesde transfert radiatif mettant en jeu des plasmas denses et multicomposantes. On metl'accent sur les traitements Monte-Carlo et deterministe, respectivement.

Abstract - This survey is devoted to several methods currently employed in the computationof radiative transfer taking place within dense and multicomponent fully ionized plasmas.We lay our emphazis on Monte-Carlo and deterministic treatments.

The radiative transfer phenomena describe the time evolution of theradiative intensity of the photons and the material temperature in a movingmaterial; thus these two quantities are coupled with the classicalhydrodynamical quantities (pressure, density, material velocity). But in usalnumerical codes, the radiative phenomena and the hydrodynamical phenomena arehandled in a separate way. So in this paper we consider only the numericaltreatment of the "radiative part'' of the equations of radiation hydrodynamics(for the statement of the whole set of equations of radiation hydrodynamics,see Mihalas-Mihalas [1] , Pomraning [2] , Buchler [3] , Munier-Waever [4] ) .

For the shake of simplicity, we will emphasize the principle of the dif-ferent numerical methods on a simplified form of these equations, in which themain numerical difficulties appear and may be analyzed.

Now, let us describe the basic model, we deal with through this paper :the material is assumed to be motionless, the photons moved linearly in aspatial domain D which is purely absorbing. So let us state the photontransport equation and the energy balance equation satisfied by the radiativeintensity u (t, x, D , v ) of the photons at time t and position x, withdirection D and frequency v , and by the material temperature T (t, ) . Here xbelongs to a domain D of R ~ , to the unit sphere S of R~ and v to R+, but bysymmetry reason x may range in a domain D of R~ (this is the two dimensionalcases) or in a domain D of ~1 (one dimensional case) :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988737
http://www.edpsciences.org/http://dx.doi.org/10.1051/jphyscol:1988737http://dx.doi.org/10.1051/jphyscol:1988737http://www.edpsciences.org/


2/16

C7-302

-where : u (v)JOURNAL DE PHYSIQUE

stands for u (fi,~)R4 r

c is the light speedk = k ~ ( x ) s the absorption coefficientR V (T) is the classical Planck function (up to the multiplicative

coefficient 4s)e (T) is the specific internal energyQ is the external source of energy

Of course, to have a well posed problem, it is necessary to give an initialcondition and a boundary condition for the radiative energy density and aninitial condition for the temperature. For the existence of solutions of sucha system, one can see for instance Golse-Perthame [5]. Now let us make somepreliminary remarks.

. It is well know that the physical behavior of the solution (u, T) isvery different in a transparent medium where the mean free path (that is tosay k$x)-l) is not small with respect to the characteristic size of the domainand in an opaque medium where it is small with respect to this characteristicsize. Let us recall that in such a case, u (t, x, 0 , v ) may be well approxi-mated by 13" (T (t,x)) where T solves the non linear diffusion equation (socalled Rosseland's equation)!

with appropriated initial and boundary conditions, where a is a radiative

constant (such that J +m BV(T)~V a c ~ ~ )nd o~ is defined by :0

(A 1.ot of papers have been written on this diffusion approximation sinceRosseland [6] . See for instance, Larsen-Radham-Pomraning [7] orj from a mathe-matical point of vue,Bardos-Golse-Perthame-Sentis [8].

So the characteristic time of the evolution of the phenomena is quite dif-ferent in the opaque media and in the transparent media, thus one can thinkthat some difficulties may be encountered in numerical solutions on a domain.v~here ransparent and opaque media are placed side by side.


3/16

. On th e o th e r hand it may be se e n t h a t t h e t e m pe r a tu r e a nd th e r a d i a t iv ei n t e n s i t y a r e alw ay s s t r o n g l y c o up l ed , t h u s it i s n e ce ss ar y t o e v a l u a t e , a te a c h t im e s t e p , t h e t e m p e r a t u r e T i n t h e r i g h t hand s i d e o f ( 1 ) by t a k i ng in -t o a cc o un t t h e e n er gy b a l an c e e q u a t i o n ( 2 ) .

So we s e e t h a t a n o t h e r d i f f i c u l t y comes fro m t h e n on l i n e a r i t y o f t h ef u n c t i o n r T- 3v ( T ) and T- ( T ) .

. Now i n two d im e ns iona l c a s e s , t h e r a d i a t iv e i n t e n s i ty u depe nds ( be yondt h e t i m e ) on 5 v a r i a b le s , t h a t i s t o say : 2 s p a t i a l v a r ia b le s ; 2 a ngu la r va -r i a b l e s a n d 1 f re q ue n cy v a r i a b l e . T h i s i s t h e r ea so n o f o t h e r n um e ri ca l d i f f i -c u l t i e s a s we s h a l l s e e f o r d e t e r m i n i s t i c sch em es.

. I n a l l t h e n u m e r ic a l meth od s o ne b e g i n s by m aking a t i m e d i s c r e t i s a t i o nby t h e f o l l o w i n g way. F or t h e s h a ke o f s i m p l i c i t y l e t us d e n o t e t h e t im e s t e pby (0, A t ) . We de no te by To a nd u" t he va lue o f T an d u a t t h e b e g i n n in g o fth e t im e s t e p . S o we ha ve t o so l ve th e f o l lowing sys te m :

w here T s a t i s i f i e s :

O wing t o a l l t h e d i f f i c u l t i e s ab ov e-m en tio nn ed , a l o t of d i f f e r e n t nu-m e r i c a l m ethods ha ve be en e l a b or a te d wi th d i f f e r e n t a ppr ox im a t ions . The a im o ft h i s p ap er i s n o t t o d e s c r i b e a l l t h e s e m eth ods b u t o n l y some o f t hem i n o r d e rt o show some d i f f e r e n t p o i n t s o f v ie w. I n t h e f i r s t p a r t , w e d e s c r i b e a M onte-3 r l o m ethod c o u pl ed w i t h a d i f f u s i o n m ethod, an d i n t h e s ec on d p a r t ,w e d e s-z r ib e some de te r m in i s t i c m e thods.

PART 1 - A MONTE CARJA METHOD

I t i s we l l known th a t t he M onte -C arlo m ethods a r e s u i t a b l e f o r so lv ingt r a ns po r t e qua t ions i n med ia where th e mean f r e e pa t hs a r e no t ve r y sm a l l . S oa n a t u r a l i d e a (o n w hich a r e ba se d a l o t o f p r o du c ti o n c o d es ) i s t o s o l v ee q u a t i o n s ( 4 ) and ( 5 ) by a Monte Ca rlo method i n th e non opaque media and t os o l v e t h e R o ss el an d e q u a t i o n ( 3 ) i n t h e o paq ue m ed ia, by a c l a s s i c a l f i n i t ee le m en t ( o r f i n i t e d i f f e r e n c e ) schem e.


4/16

C7-304 JOURNAL DE PHYSIQUE

To c o u p le t h e two me th od s, t h e r e a r e a l o t o f d i f f e r e n t w ays, w hic h a r eq u i t e t e c h n i c a l ( an d w hi ch a r e n o t d e s c r ib e d h e r e ) . L e t u s f o c c us e o u r s e l v e so n t h e mo st u s u a l M o nt e- Ca rl o m eth od t o s o l v e e q u a t i o n s ( 4 ) - (5), t h e s o -c a l-l e d F l e c k ' s m eth od [9] .

F i r s t of a l l , l e t u s make a f re q ue n cy d i s c r e t i s a t i o n . The f r eq u en c y s e t[o.+ -1 i s d i v i d ed i n t o N i n t e r v a l l s ( o r g ro up s) vO= 0 , 31, Q 2 .. ,pNF o r t h e f r eq u en c y v a r i a b l e 3 , we s u b s t i t u t e t h e d i s c r e t e v a r i a b l e n an dw e d e f i n e :

. k n t h e a v e r a g e v a l u e o f k o v e r t h e " gr ou p" n t h a t i s t o s a y ( '3n-1,3,)

. 13, ( 8 ) t h e i n t e g r a l o f t h e P l an ck f u n c t i o n o v er t h e g ro up n a t t h et e m p e r a t u r e ( e / a ) 1 / 4 t h e n 1 3, = c en. un t h e i n t e g r a l o f t h e i n t e n s i t y u o n t h e g r o u p n.T hen e q u a t i o n s ( 4 ) a n d ( 5 ) may b e w r i t t e n i n t h e f o l l o w i n g fo rm :

1 u n ( 0 , .) = u OlA t7) e ( e ( a t ) ) - e ( e O ) = - [kn En - k n n n ( e l ] d t0 ' n

Now, t o a v o i d i n s t a b i l i t y , t h e p r i n c i p l e o f t h e method i s t o h a v e a goode v a l u a t i o n o f t h e t e m p er a t ur e w hich a p p e ar s i n t h e r i g h t h an d s i d e o f ( 6 )an d (7), h a t i s t o s a y a n e v a lu a t io n o f e w h i c h i s c l os e d t o t h e v a l u e o f t h ek e m pe r at u re " 8 a t t h e e nd o f t h e t i m e s t e p .

To do t h i s , w e a p p r o x i m a t e 8 , ( 0 ) by the f o l l o w i n g w ay :

w h e r e 0 s a t i s f i e s a l i n e a r i z e d f o r m ,o f ( 7 ) :

w i t h k p = c e O


5/16

Thus on the time interval1 (0, A t) we have to solve the so called Fleck'sequation :

-here a pseudo-scattering term (kn an (6 ) E km u is substitutedmfor a part of the emission term.

It is not difficult to solve this equation by a Monte Carlo method even ina 2 dimensionnal geometry with an arbitrary Lagrangian mesh. The particles areemitted from the source (including voiune emission and initial radiationenergy) : they move linearly (so intersection of the trajectories and the ed-ges of the cells are to be calculated for each particle) with weights whichdecrease exponentially (in order to take into account the absorption term) ;they undergo collisions with a probability of exponential type whose mean freepath is :

After the collisions, the direction and the frequency of the particles arechanged (in order to modelize the scattering operator).

It may be easely seen that this tracking of the particles is praticableonly if the mean free path dn between two collisions is not too small, thatis to say if the opacities k, are not too large.

If it is not the case (especially in the opaque media) the method has tcbe modified by a "Random Walk procedure" whose aim is to avoid very complextrackings of the particles : a jump is used as a substitute for the trajectoryof a particle which undergoes a large number of collisions in a cell. Some pa-pers has been written the emphasize the characteristics of this "jump method"see for instance

After the solution of the Fleck's equation (what is to say after the trac-king of the particles), one has to consider the energy balance equation (7)which has been modified to take into account the linearisation of 8, ( 0 ) by


6/16

JOURNAL DE PHYSIQUE

Pn ( e O ) 8 , t h a t i s t o s a y :8

-I n t h i s e q u a ti o n t h e t er m k m um ( t ) d t mu st b e e s t im a te d by a wayw hic h i s a p p r o p r i a t e d t o t h e M on te -C arlo m ethod, i n f a c t t h i s t e rm i s r e l a t e dt o t h e a b s o r b e d r a d i a t i o n e n e rg y a n d th e r i g h t h an d s id e of C10) may b ei n t e r p r e t e d a s t h e d i f f e r e n c e o f t h e a b so rb ed e n er gy a nd t h e e m i t t e d e ne rg y.

The F l e c k ' s d i s c r e t i s a t i o n of t h e r a d i a t i v e t r a n s f e r e q u a t i o n has beena n a ly z e d f ro m t h e p o i n t o f v ie w o f t h e s t a b i l i t y i n t h e p a p e r of L ar se n-w e r c i e r [10]

To summarize, we can c la im t h a t Monte-Carlo method i s w e l l s u i t e d f o rs o l v i n g r a d i a t i v e t r a n s f e r e q u a t i o n s i n t r a n s p a r e n t m edia b u t i n o paque mediait i s n o t p ra t i c a b l e u n l e s s it i s a c c e l e r a t e d by a Random W a l k p r o c e d u r e(whose accuracy i s n o t s o goo d a s t h e on e o f t h e F l e c k ' s m e th o d ) .

On the o th e r hand , t h i s me thod ha s some d isad van tage s :

- it i s h i g h l y C.P.U. t ime consumming ( e v e n i f t h e t r a c k in g o f t h ep a r t i c l e s i s v e c to r i ze d ),- o f c o u r s e i t e x i s t s s t a t i s c a l f lu c t u a t io n s i n t h e r e su l ts ,

- cou pl ing be tween th e Monte-Carlo method in t he t r a ns pa re n t media and th ed i f f u s io n meth od i n t h e o pa qu e med ia i s n o t s imp le a nd , may b e , n o t v e r ya c c u r a t e : th e nu mer ica l r e s u l t depends ve ry much on th e way used f o rt h i s c o up li ng .

So , fo r a long t im e , o th e r me thods have been s tu d i ed which used the samesc heme bo o th i n t h e t r a n s p a r e n t med ia a nd th e o p a qu e med ia . A l l these me thodsa r e d e t e r m i n i s t i c .

PART 2 - DETERMINISTIC METHODSB r i e f l y s p e a k i n g , t h e r e a r e tw o k i n d o f d e t e r m i n i s t i c m etho ds :

- t h e metho ds w hic h a r e b a s ed o n th e d i f f u s io n a p p r o x ima t io n o f t h et r a n s p o r t e q u a t i o n , w hic h a r e c a l l e d P1 m e th od s, o r m u l t i f r e q u e n c yd i f f u s i o n c a l c u l a t i o n s- t h e - m e t h o d s w hich d e a l w i t h t h e t r a n s p o r t e q u a t i o n s w i t h s p a t i a l an dangula r dependance .


7/16

1 - P1 - METHODST he g e n e r a l p h i l o s o p h y o f t h e s e me th od s i s t o rem ark t h a t , v e r y o f t e n , t h e

r a d i a t i v e i n t e n s i t y d oe s n o t d ep en d v e ry much on t h e a n g u l a r v a r i a b l e ( f o ri n s t a n c e i n t h e o pa qu e m ed ia , a nd i n t h e t r a n s p a r e n t m e dia when t h e t i m eb e h av i or o f t h e s o l u t i o n i s q u i t e s t a t i o n a r y ) . Then it i s assum ed t h a t t h i si n t e n s i t y i s a l i n e a r f u nc t io n o f t h e a n gu l ar v a r i a b l e t h a t i s t o s a y :

( 1 1 ) u ( t , x , n , v ) = c y , ( t , X , V ) + 3 n~ ( t , x , v )

where y, ( t , x , v ) = 1 v ( t , x).C

A f t e r w a r d s it may s een t h a t :

1L e t u s d e n o t e B v ( T ) = - 13v ( T )C

L e t u s w r i t e t h e f i r s t moment e q u a t i o n d e r i v e d f ro m ( 4 ) :

aFBy n e g l e c t i n g w i t h e s p e c t t o ckv F , w e f i n d :

a t

hen, from t h e i n t e g r a l ( o ve r S ) of t h e e q u a t io n ( 4 1 , we g e t :

And t h e e n e r g y b a l a n c e e q u a t io n . m a y b e w r i t t e n i n t h e f o l l o w i n g f or m :

A l o t o f w o rk s h a v e b e e n p e rf o rm e d , t o m o di fy t h e f o r m u l a t i o n ( 1 3 ) - ( 1 4 )i n o r d e r t o u se it i n t h e t r a n s p a r e n t m edia. The a im o f t h e s e w o rk s i s t om o di fy t h e e q u a t i o n (12), by t h e i n t r o d u c t i o n o f " Fl ux l i m i t e r s " o r " Ed di ng to nf a c t o r s " .


8/16


The Eddington factor is a factor u which is evaluated by some physical ornumerical consideration (and which is equal to 1/3 in the opaque media) andwhich is used to modify the equation (12) :

Such methods are usually used since the VERA code [14]. See also [15] and[16]. A very good survey of all the works on this subject has been publishedin Pomraning 1171 or [18].

Another way to deal with this problem, is to evaluate the "Flux cor-rection"

- C a( PG = (F -- (which is non zero only in the transparent media) by a3k, a x

particle method. In fact it is well known that particle methods are prac-ticable only if the mean free path is not too small that is to say in thetransparent media (see Sentis [19] ) .

Now let us go back to the system (13) (14).

Let us make the frequency discretisation by the same way as in Part I. Letus define :

kn the arithmetic average of k on the group nan the harmonic average of k on the group n

B n(.) the integral of B y ( . over the group ncnD, Cn the integral of yo. 'f) over the group n at the beginning and

the end of the time step (o,At)

To and T the temperature at the beginning and the end of the timestep.

Then, after a classical implicit time-discretisation, it may be seen that onehas to consider,the following system (instead of system (13) (14)):

c a t ax 30, ax

This is a system on N linear diffusion equations which are coupled alltogether through a non linear energy balance equation. In two dimensional


9/16

problems, a direct method based on classical linearisations seems not to bevery practicable.

Several methods have been developped to solve this system. Let usmention :

. the "multifrequency-grey" method (see [14] [20]). the "partial temperature" method (see [21])But recently a method based upon the "diffusion synthetic acceleration" hasbeen studied (the first publications related to this method are 1221 and[23]). It is based upon the classical method of diffusion syntheticacceleration for linear transport equation (see the references of [22] ) andupon the Newton method to solve non linear systems.

At each Newton iteration, one solves the N linear diffusion equationsusing an acceleration scheme.There exist three ways to linearize the Planckfunction around To :

The linearization a) has been introduced in [22] , the linearization b) in'233 (in a more general framework) and c) in [24]. Remark that a) is the samelinearization than the one introduced by Fleck [9] for the Monte Carlomethod. The linearizations a) and b) have been compared by Larsen [26]. From anumerical point of view, all the three linearizations yield very similarresults.

Now let us describe an algorithmus based upon the "diffusion syntheticacceleration" (D.S.A.) method and the linearization a). Notice that a lot ofalgorithms based upon the same method may be developped.

Let e0 = a~'4 and = aT4. Let us write R, ( e), e(B ) instead of Bn(T)and e(T). Let bo, = Bn (eo)/Bo, T = cat, EO =(&) ( eo).

a e

Iteration 1

At the first iteration of the Newton method, the theoritical rnultigroup


10/16

JOURNAL DE PHYSIQUE

s ys te m t o s o l v e i s t h e f o l l o w i n g o ne :

( 1 8 ) ~ " 6 6" 8" - r 1 k m Y m + r k p 6 = Q om

On c ou rs e it i s n o t n ec e ss a ry t o s o l v e e x a c t l y t h i s sys tem . I n f a c t , ones o l v e s f i r s t a s ys te m w he re a l l f r e q ue n c y gr o up s a r e de c ou pl ed ; a f t e r w a r d so n e u s e s a " gr ey d i f f u s i o n " ( 1 ) a c c e l e r a t i o n e q u at io n t o t a k e i n t o ac c ou nt t h ec o u li n g between a l l t h e f re qu e nc ie s . T hus t h e f i r s t i t e r a t i o n i s d i v i d ed i n t o3 s t e p s .a) Multigroup step . L e t t o b e t h e s o l u t i o n of t h e e q u a t i o n ( 1 7 ) b u t%t h e r i g h t hand s i d e i s changed by :

b ) T h i s i s t h e a c c e l e r a t i o n s t e p . I n t h e p r e vi o u s s t e p t h e c o u p li n g o p e r a to rh a s b ee n n e g l e c t e d . So we add t o q1j2 a t e r m w hi ch t a k e s i n t o a c c o un t

nt h e c o up l in g t h a t i s t o s a y a t e r m o f t h e f o l l o w i n g f o r m :

w h e r e wl d o e s n o t - d ep en d o n n a n di ss u c h t h a t ( + b o n wl) i s c l o s e'nt o t h e s o l u t i o n of ( 1 7 ) . T hus, l e t w1 t o be t h e s o l u t i o n o f t h e f ol lo w in g" g r e y " e q u a t i o n .

( F o r th e d e t a i l s a nd t h e j u s t i f i c a t i o n o f t h i s p r o c e d ur e s e e [22] [23] [24]) .c ) L e t t o b e d e f i n ed by :

n

( 1 ) T h i s e q u a t io n i s c a l l e d g r e y b e c a a s e t h e s o l u t i o n d oe s n o t d ep en d on t h ef r eq u e nc y v a r i a b l e n


11/16

a nd e1 d e f i n e d by :

Iteration 2One l i n e a r i z e s B, ( . an d e ( . a r o u n d t h e t e m p e r a t u r e e 1 :

i Bn (0 ' )where bn = e 1a ee ( 0 ) e ( e l ) + I ( - 0 ' 9 where E ' ( - 1 ( e l )a e

The t h e o r e t i c a l m u lt ig r ou p s y st em t o s o l v e i s t h e f ol lo w in g . :.

1w i t h K1 = Z km b,mThe i t e r a t i o n i s d i vi d ed i n t o 3 s t e p s a s b e f o r e .

a ) L e t b e t h e s o l u t i o n o f t h e e q ua t i o n (201, b u t t h e r i g h t han d s i d ei s changed by :

b ) L e t w2 b e t h e s o l u t i o n o f a g r e y e q u a t i o n w h ic h i s o f t h e sam e t y p e a s(19).

c 2 and e 2 a r e d e f i n e d b y t h e same way a s b e f o r e .40A f t e r w a rd s o n e i t e r a t e s u po n c o n v er g e nc e .

To summarize, it may b e s e e n t h a t t h e D.S.A. m eth od s l e a d o n l y t o s o l v el i n e a r d i f f u s i o n e q u a t i o n s . T h at may b e e a s i l y d o n e e v en i n a t w o -d i me n si o na lL a g r a ng i a n m esh . A no t he r a dva n t a ge o f t h e s e m et hods i s t h e f o ll o wi n g : it mayb e p ro ve d t h a t i n t h e o pa qu e media t h e s o l u t i o n o bt a i ne d a f t e r o n e i t e r a t i o no f t h e s chem e i s c l o s e t o t h e r i g h t s o l u t io n , t h a t i s t o s a y t h e s o l u t i o n o ft h e R o ss e la n d d i f f u s i o n e q u a t i o n ( s e e f o r i n s t a n c e [25: o r 1261 1.


12/16

JOURNAL DE PHYSIQUE

2 - METHODS WHICH DEALS WITH TRANSPORT EQUATIONS

I t i s a n a t u r a l i d e a t o u s e t h e sam e i d e a s r e l a t e d t o t h e D.S.A. methodf o r s o l v i n g t h e N m u l t if r e q ue n c y t r a n s p o r t e q u a t i o n s ( c o u pl e d w i t h t h e e n e rg yb a l an c e e q u a t i o n ) i n s t e a d o f t h e N m u l t if r e qu e n c y d i f f u s i o n e q u a ti o n s .

F o r s o l v i n g a l i n e a r t r a n s p o r t e q ua t io n it i s u s u a l t o u s e a s t a n d a r dDSN s cheme a c c e l e r a t e d by a l i n e a r d i f f u s i o n e q u at i o n. B u t f o r t h e r a d i a t i o np ro bl em s t h e t r a n s p o r t e q u a t i o n de pe nd s on t h e f r e q u e nc y v a r i a b l e , t h u s t h ed i f f u s i o n e q u a t i o n d ep en ds a l s o o n t h e f r e q u en c y , a nd o ne h a s t o s o l v e a s y s -t e m o f N l i n e a r d i f f u s i o n e q u a t i o n s wh ich a re c o u p l e d a l l t o g e t h e r t h ro u gh t h en on l i n e a r e n e r gy b a l a n c e e q u a t i o n . S o we h a v e tw o i t e r a t i o n s ch em e s w hi cho v e r l a p o ne w i t h t h e o t h e r ( t h e s e i d e a s a r e d e ve lo pp ed i n [23] , [28], [29] ) .

F o r t h e i m p l e m e n t a t i o n o f s u c h m e th o ds it is n e c e s s a r y t o h a v e a n im pro vedDSN s ch em e f o r s o l v i n g e a c h t r a n s p o r t e q u a t i o n .

F i r s t , t h e s t a n d a r d DSN s ch em e m us t b e m o d i f ie d t o y i e l d a lw a y s p o s i t i v es o l u t i o n s w i t h t h e i n t r o d u c t i o n o f s e t - t o -z e ro f i xu p ' f o r i n s ta n c e : t h a t l e a d st o m odify t h e d i f f u s i o n s y n t h e t i c e q u a t i on i n o r d e r t o b e c o n s i s t e n t w i t h t h i sf i x u p ( a s u r v e y o n t h e s e m e t h o d s may be f o u n d , f o r i n s t a n c e i n [27] ) .

I t i s a l s o n ec es sa ry t o t a k e c a r e t o t h e s p a t i a l d i s c r e t i s a t i o n o f t h e r a -d i a t i v e i n t e n s i t y which m u s t b e e v a l u a t e d a t t h e same p o i n tS t h a n t h e m a t e r i a lt e mp e ra t ur e i n o r d e r t o a vo id d i f f i c u l t i e s f o r s o l v i ng t h e e ne rg y b a la n cee q u a t i o n .

T h es e f a c t s l e a d t o t h i n k t h a t t h e s e DSN s ch em es w i t h D.S.A. method may bep r a c t i c a b l e f o r pr o bl em s i n a o ne d i m e n s io n a l g e om e tr y o r i n a t wo -d im en si on alr e c t a n g u l a r g e o m e tr y b u t n o t o n a t wo d i m e n s i o n a l L a g r a n g i a n m esh ( w h e re a l o t

,o f o t h e r d i f f i c u l t i e s a r e en co un te re d , f o r i n s t a n c e t h e p ro blem o f l i g h t i n g o ft h e c e l l s , s e e f o r i n s t a n c e [30] 1.

. A n o t h er c l a s s o f m et ho ds , b a s e d up on F e a u t r i e r 8 s i d e a s [31] a r e s t u d i e df o r a l o ng ti m e . The b a s i c i d e a i s t o s o l v e a l i n e a r t r a n s p o r t e q u a t i on byu s i n g t h e e v e n -p a r it y e q u a t i o n , t h a t i s t o s a y t h e e q ua t i o n s a t i s f i e d b y t h es ym m et ri c a v e ra g e o f r a d i a t i v e i n t e n s i t y :

(where G b el o ng s t o t h e h a l f s p h e re ) .T h e se m eth od s a r e u s e d i n n e u t r o n i c s ( s e e M i l l e r i3 2, o r V er wa er de [33]

f o r i n s t a n c e ) a nd i n r a d i a t i v e t r a n s f e r ( s e e M i ha la s [34] f o r a s u r v ey o n t h i ss u b j e c t ) . T he ob v i o us a d v a nt a g es o f t h i s t e c h n iq u e i s t h a t t h e D i f f u s i o n Syn-


13/16

thetic Acceleration equation has the same structure than the angular even-parity equation. But as in the DSN method, one has to take care to the spatialdiscretisation : The material temperature has to evaluated at the same pointsthan the radiative intensity.

To the author's knowledge, this method has been implemented for radiativetransfer problems only in one-dimensional codes with planar or spherical sym-metry (where the solution of even-parity equation leads to the inversion of a5-diagonal matrix). In fact, it is much more difficult to implement this me-thod in two dimensional geometry with cylindrical symmetry (where that leadsto the inversion of a 7-diagonal matrix).

To conclude, let us recall that the solutions of the radiative transferproblems may be very stiff, the material temperature may have important dis-continuities with respect to the spatial variable and the radiative intensitymay range over a large number of magnitude with a small variation of the fre-quency variable. It is one of the reason of the difficulties encountered inactual numerical schemes. At last, one can say that a lot of improvements canbe brought to the above described methods but some tools, such as the dif-fusion synthetic acceleration method, are essential for efficient and accuratecomputations.

REFERENCES

General

1. D. Mihalas and B.W. Mihalas. Foundations of Radiation Hydrodynamics,Oxford University Press New-York, 1984.

2. G.C. Pomraning. The equations of Radiation Hydrodynamics, Pergamon,Oxford (1973).

3. J.R. Buchler. Radiation Transfer in the fluid frame. J. Quant. Spectros.Radiat. Transfer 30 (1983) p. 395-408.

4. A. Munier - R. Waever. Radiation transfer in the fluid frame : acovariant formulation (I and 11) Computer Physics Reports 3 (1986)p.125-164.

5. F. Golse - B. Perthame. Generalized solutions of the radiative transferequations in a singular case. Commun. Math. Phys. 106 (1986) p.211-239.


14/16


6. S. Rosseland, Handbuck d&r Astrophysics 3 (I) p. 433 (1930).

7. E.W. Larsen, G.C. Pomraning, V.C. Badham. Asymptotic Analysis ofradiative transfer problems.J. Quant. Spectros. Radiative transfer 29 (1983) p. 285-310.

8. C. Bardos, F. Golse, B. Perthame, R. Sentis. The non accretive RadiativeTransfer equations : existence and Rosseland approximation. J. ofFunctional Analysis 77 (1988), p.434-460.

MONTE-CARLO Method

9. J.A. Fleck, J.D. Cummings. An Implicit Monte-Carlo scheme forcalculating. . non linear Radiation Transport, J. Comp. Physics 8 (1971)p.313.

10. E.W. Larsen - B. Mercier. Analysis of a Monte-Carlo Method for non linearRadiative Transfer. J. Comp. Physics 71 (1987) p.50.

11. J.A. Fleck - E.H. Canfield. A random walk procedure for improving thecomputational efficiency, J. Comp. Physics 54 (1984) p.508-523.

12. J. Giorla - R. Sentis. A Random Walk method for solving RadiativeTransfer equations.J. Comp. Physics 70 (1987) p.145-165.

13. J.E. Lynch 2 ; roceedings of the Joint LANL-CEA meeting "Monte-CarloMethods and Applications Cadarache Castle" - France 1985. Ed. by Alcouffeand al. Lectures Notes in Physics Vol. 240. (springer Verlag - Berlin -1985) p.106-115.

Deterministic Methods

14. B E . Freeman, L.E. Hauser and al. The VERA code, one dimensionalradiative Hydrodynamic Program, Vol. 1 - D.A.S.A. report 2135, SystemScience and Software, La Jolla (1968).

15. G.C. Pomraning - An Extension of the Eddington Approximation J. QuantSpectros. Rad Transfer 9 (1969) p.407-422.

16. C.D. Levermore - Relating Eddington factors to flux limiters J. QuantSpectros. Rad Transfer 31 (1984) p.149-160.

17. G.C. Pomraning. A comparison of various Flux limiters and Eddingtonfactors". Lawrence Livermore Nat.Lab. Report UCID-19220 (1981).


15/16

18. G.C. Pomraning. Radiation Hydrodynamics. L80s Alamos Report LA-UR-82-2625(1982).

19. R. Sentis. ~6solution um6rique dlQquations de transport avec correctionparticulaire (unpublished).

20. C.M. Lund and J.R. Wilson. Lawrence Livermore Nat. Lab. Report UCRL 84678(1980).

21. A.I. Shestakov, J.A. Harte, D.S. Kershaw. Solution of the DiffusionEquation by Finite Elements in Lagrangian Hydrodynamic Codes. J. Comput.Physics 76 (1988) p.385.

22. J.E. Morel - E.W. Larsen - M.K. MatzenA synthetic Acceleration scheme for Radiative Diffusion Calculation.J. Quant Spect. Rad. Transfer 34 (1985) p.243-261.

23. R.E. Alcouffe - B.A. Clark - E.W. Larsen. The synthetic acceleration oftransport iterations, in Multiple Time Scale (J. Brackbill - B. Cohened. Academic Press, New-York (1985)

24. R. Sentis - M. Parmentier. Mbthode multigroupe Directe implicite pour larbsolution du transfert radiatif. Note CEA-N-2473 (VilleneuveSaint-Georges France) (1986).

25. E.W. Larsen. Stability of time differenced radiative Transfer equations,Proceedings of the "Code Developers ~eetin~: an Diego (nov. 84).

26. R. Sentis. Multigroup Direct Methods for solving Radiative Transferequations. To be published.

27. E.W. Larsen. Diffusion synthetic acceleration methods for discreteordinates equations. Proceedings of the A.N. S. meeting "Advances inReactor Computations" Salt Lake City (1983).

28. J. Honrubia. A multigroup radiation hydrodynamics algorithm for theanalysis of ion beam energy conversion (in these proceedings).

29. B.A. Clark. Non linear D.S.A. difference schemes for radiative transferapplications. tos Alamos Report.

30. T.R. Hill - R.R. Patermoster. Two dimensional spatial discretizationmethods on a Lagrangian mesh. Los Alamos Report (1982).

31. P. Feautrier. Compte-rendus Ac. Sciences Paris 258 (1964).


16/16


32. E.E. Lewis. Finite Element Approximation t o the Even-parity TransportEquation" Adv. Nucl. Sci. Tech. 13 (1981).

33. D. Verwaerde. Rdsolution de l'dquation du transport par la methode duFlux pair. Note CEA-N-2352 (1983).

34. D , Mihalas. The Computation of Radiation Transport Using FeautrierVariables. J. Comp. Physics 57 (1985) p.1.

R. Sentis- A Short Survey of Numerical Methods for Radiative Transfer Problems

Documents

Transcript of R. Sentis- A Short Survey of Numerical Methods for Radiative Transfer Problems