Numerische Lösungstechniken der Strahlungstransportgleichung mittels verallgemeinerter mittlerer Intensitäten

Computing 54, 27-38 (1995) ~ [ ~ [ i [ ' l ~

�9 Springer-Verlag 1995 Printed in Austria

A Generalized Mean Intensity Approach for the Numerical Solution of the Radiative Transfer Equation

S. Turek, Heidelberg

Received January 15, 1994; revised July 4, 1994

Abstract - - Zusammenfassung

A Generalized Mean Intensity Approach for the Numerical Solution of the Radiative Transfer Equation. In [7] we proposed a general numerical approach to the (linear) radiative transfer equation which resulted in a high-dimensional linear system of equations. Using the concept of the generalized mean intensity, the dimension of the system can be drastically diminished, without losing any information. Additionally, the corresponding system matrices are positive definite under appropriate conditions on the choice of the discrete ordinates and, therefore, the classical conjugate gradient-iteration (CG) is converging. In connection with local preconditioners, we develop robust and efficient methods of conjugate gradient type which are superior to the classical approximate A-iteration, but with about the same numerical effort. For some numerical tests, which simulate the astrophysically interesting case of radiation of stars in dust clouds, we compare the methods derived and give some examples for their efficiency.

Key words: Radiative transfer equation, mean intensity, nonsymmetric CG-variants, astrophysics

Numerische L6sungstechniken der Strahlungstransportgleichung mittels verallgemeinerter mittlerer Intensitiiten. In [7] ffihrten wir einen allgemeinen Zugang zur numerischen L6sung der (linearen) Strahlungstransportgleichung ein, der aber zu sehr hoch dimensionierten Gleichungssystemen fiihrte. Indem wir das Konzept der verallgemeinerten mittleren Intensitgt verwenden, kann die Dimension drastisch verringert werden, ohne dabei Information zu verlieren. Darfiberhinaus werden die Systemmatrizen unter geeigneten Bedingungen an die Wahl der diskreten Ordinaten positiv definit, und das klassische konjugierte Gradientenverfahren (CG) ffihrt zu Konvergenz. In Verbindung mit lokalen Vorkonditionierungstechniken k6nnen wir robuste und effiziente Methoden von konjugierten Gradiententyp konstruieren, die sich bei gleichem numerischen Aufwand der klassischen approximativen A-Iteration als iiberlegen erweisen. In numerischen Tests, die den astrophysikalisch interessanten Fall yon Sternstrahlung in Staubwolken simulieren, vergleichen wir die verschiedenen hergeleiteten Methoden und geben Beispiele flit ihre numerische Effizienz.

1. Introduction

In this paper we consider a new numerical approach to the (frequency decou- pled) linear radiative transfer equation, formulated for the specific intensities I(x,w) as

n~o" VxI+ K(x,m)I=~sR(X,W,w')I(x,w') do)' +f(x,(o), in ~), (1)

u(x,w) =g,o(x) on F~. (2)

28 S. T u r e k

Here, ~) and F~- are defined as

g~: = ~2 • S with ~2 c R 3 and the unit sphere S c R 2,

F,~:= {x~ 3 a l n x . n w <0}, (3)

where n x represents the outward normal unit vector to 3S2 at x ~ 3~2 and n w �9 Vx the directional derivative for the direction vector corresponding to w. We assume the standard product expansion ([4]) for the scattering phase function R

R =

Our first numerical approach to this system of integro-differential equations was done in [7], where we ended with two main problems:

1) The high dimension (3D in space, 2D for the ordinates) of the problem. 2) The lack of convergence theory for the nonsymmetric CG variants.

In this paper we additionally assume

A = A(x), P = P ( o ; ) , ~sP(co')dco' = 1, (4)

which is quite common in astrophysics, e.g. P = 1/4~- (see also [3], [4], [8]). Under this additional assumption we derive a solution method with (at least) the same numerical convergence properties as in [7]. However, our new approach requires a much smaller amount of storage which is even independent of the number of discretization points on S, and we are able to show convergence for our linear solvers in (almost) all cases.

2. Space and Ordinate Discretizations

We discretize the unit sphere S by a set of surface points ~o m, m = 1, . . . , M, and replace the integral expression on the right hand side by an approximate quadrature formula

M

~ P(w')I(x,w')dco'= E cm(p)I(x,~ (5) ~ X

m = l

with quadrature weights cm(p). A standard approach in astrophysics is to introduce spherical coordinates which yields

fsP(W')I(x,co')doa'= fo~s sin q~' d~'. (6)

Approximating separately both one-dimensional integral expressions, e.g. by the trapezoidal rule, leads to

L K

/ = 1 k = l

Mean Intensity Approach for the Numerical Solution of the Radiative Transfer Equation 29

with quadrature weights r

c'h(P) = 2-Z- sin 9' P(0 h, 9').

In this case, we get for the direction vector that there holds

n,o,k = (sin 91 cos O k, sin 91 sin 0 h, cos 9l) r.

(8)

(9)

Following this semi-discretization, we have to solve the following coupled system of PDE's

M no)m'ITxlm(x ) q-t<m(x)Im(x) = A ( X ) E cn(p)In(x) +fm(x),

n=l m = l , . . . , M ,

(10)

with the semi-discrete approximations

Im(X) =I(x,ogm), Kin(X) = K(X,C0m), fro(X) =f(x,c-om).

Next, we introduce the generalized mean intensity J(x) by M

J(x) : = E c"(V)I~ (11) n= l

which leads to the system

no, m ' ~ I ' ~ ( X ) + , ~ ' ( X ) I m ( x ) = a ( X ) l ( x ) + f m ( x ) , m = l . . . . ,M. (12)

To continue we need a spatial discretization. Let Th z Ih m be a discrete version of the left hand side in the domain /2

T~I~ ~ n,o.," ITxlm( x) q- Krn( x)Im( X), (13)

and analogously LhJ h of A(x)J(x), respectively, fh m of fro(x).

An example is the finite element approximation by rotated bilinear elements in connection with an appropriate upwinding strategy (for more details see [7]). Using this approach, Eq. (10) is multiplied by test functions of our trial space Sh(g2), and integrated over /2. Then, I~ is the coefficient vector of the finite element solution function corresponding to Im(X), Z~ n the discretization matrix due to the differential operator no, re" Vx+ Kin(x) and L h the mass matrix corresponding to the coefficient function A(x). The reason for using this finite element discretization is its high flexibility in combination with accuracy (see also [6]) and the fact that the mass matrix L h is a diagonal matrix if an appropriate mass lumping is performed. Additionally the upwinding procedure guarantees stiffness matrices Th ~ which are always lower triangular matrices if a corresponding renumbering is done, and this process works independently for all geometrics and meshes. These points are very essential for the fast implementa- tion of the following numerical solution methods.

Other possibilities are discussed in [1]. One of our favourite discretization schemes is a finite difference method which allows first or even second order

30 S. Turek

discretization of the spatial derivative on very general meshes. These methods always lead to lower triangular matrices, too, and will be discussed more precisely in a forthcoming paper.

Following the steps performed so far our discrete form of the system (12) reads:

T ~ I ~ = L h J h +fh m, m = 1 , . . . ,M, (14)

respectively,

I~ = ( T ~ n ) - l ( L h J h +f~") , m = 1 , . . . , M . (15)

Multiplying with c m ( p ) and summing up for all m leads to M M

E C m ( p ) I ~ = E C m ( p ) ( T ~ " ) - l ( L h J h + f ~ ' ) , (16) m=l m = l

respectively,

with

rh=

Jh = ThLhdh + Fh (17)

M M Y'. c m ( p ) ( T ~ n ) -1 a n d Fh= ~.~ c m ( p ) ( T f f ~ ) - l f ~ n. ( 1 8 )

m = l m = l

We end up with the (well known) discrete mean intensity formulation (see [2])

( I h - T h L h ) J h = Fh, (19)

where I h denotes the identity matrix. The size of this discrete system depends for an arbitrary number of discrete quadrature points for the ordinates on the spatial discretization only. Nevertheless, the numerical effort for the solution process is about the same as for the method proposed in [7]. The same number of transport problems with stiffness matrices Th m has to be solved for each discrete ordinate, but in a sequential way such that partial results may be added. Once having calculated the mean intensity Jh, also the specific intensities Ih m are determined and can be computed in a single postprocessing step by solving Eq. (14) for the desired indices m.

3. Numerical Solution Methods

Our aim is to solve the linear system A h J h = F h with system matrix

A n = I h - T h L h . (20)

Since T h is the weighted sum of many inverse transport matrices, A h cannot be given explicitly and iterative methods have to be used. The common (physical) assumptions

K(x,o,) _> ,~(x) _> 0 (21)

and the use of upwinding discretizations which ensure that the transport matrices Tff are of positive type, guarantee that A n is a definite matrix with

0 < cond(I h - ThLh) _< 1. (22)

Mean Intensity Approach for the Numerical Solution of the Radiative Transfer Equation 31

Therefore, the simple weighted Richardson iteration

J<nn+l)=J<hn>--O'h(AhJ(hn)--Fh), 0 < o r h < 2 (23)

is converging (see also [7]). It is obvious that the special case ~r h = 1 corresponds to the well known approximate A-iteration which is mainly in use by astrophysi- cists (see for instance [3], [5], [8], [10]). This well known, but usually slow Richardson method, especially in the case of high extinction K and scattering values, A, can be accelerated by preconditioning

j(h,+ 1)= j(h,) _ CrhChl( Ahj~,) _ Fh ), (24)

where C h should be easy to invert and Ch a A h = I h in any sense. The problem is that we do not have an explicit representation of A h which would enable to use SOR- or ILU-type preconditioners. One simple but nevertheless successful method is to take the diagonal of A h which can be calculated due to the lower triangular form of the partial matrices Th ~ by

M

Ch = diag(Ah) = 2 cm(P)(diag(T~')) -1. (25) m = l

This approach which requires the additional numerical effort of scaling by a vector can be easily implemented and is well known in astrophysics ([3], [5], [10]) as purely local approximate A-operator. Another possibility, also known as non-local or nearest neighbour approximate A-operator, is to calculate the first subdiagonal of each (Thin) -1 and to sum it up for all m. The resulting matrix is (still) a sparse matrix, but we have to invert it in (24) which entails considerable numerical effort, since the properties of this matrix and hence the choice of the iterative solver are not clear yet. Therefore, we delay the computational results for this approach to a forthcoming paper.

In [7] we have seen that in the case of large optical opacity (K >> 1) and large scattering values (A/K ----- 1), the usual approximate A-iteration collapses and the convergence rates behave like the ratio A/K, a fact which cannot be essentially improved by a local preconditioner. A remedy proposed in [7] was to use (nonsymmetric) conjugate gradient-like methods, especially the Bi-CGSTAB (see [9]), which works much better in our case. The philosophy behind this approach is that CG-methods can be interpreted as improved Richardson- schemes where the relaxation parameters o- h are determined adaptively in each iteration step. Additionally, the analysis for the eigenvalues of the operator A h shows a strong clustering which is favourable for these schemes. But one remaining theoretical problem is that we cannot give an analytical convergence proof since no general results are known about the convergence behaviour for the Bi-CGSTAB applied to M-matrices. Here, in the mean intensity formulation and for K >__ h >__ 0, we can overcome this problem by a simple trick. We consider Eq. (17)

Jh = ThLhJh + Fh = ThL1/ZLlh/zJh + Fh. (26)

32 S. Turek

Multiplying from left with L 1/2 and substituting • = L1/2jh, 1~ h = Llh/2Fh, we obtain

fh : Lk/2ThLk/2fh + Fh, (27)

respectively,

( Ih - Llh/ZThLlh/2)fh =/?h. (28)

The resulting matrix Ah = (Ih - r l / Z T r l / % ~h *h'~h " is symmetric iff T h is symmetric which is the weighted sum of the transport matrices (Thin) -1. Here, we use a theoretical result which can be found, for instance, in [2], that T h converges for h ~ 0 to a semi-discretized operator T = T ( M ) which may be a selfadjoint operator. The criterion is that for each ordinate respectively direction o) m its counterpart with opposite direction has to be in the discrete set of ordinates {o) m, m = 1 . . . . . M}, which is always satisfied by the trapezoidal rule.

With this (not restrictive) condition on the discrete ordinates and the assump- tion that

T ~ I ~ ~ n,m" Vxlm(X) + K m ( x ) l m ( X ) for h ~ 0, (29)

our discrete operator T h converges to a selfadjoint operator T. In the case of the finite difference approach, we can establish the symmetry of Ah only (asymptotically) for small mesh sizes h, while in the finite element context this condition is satisfied for all h.

Therefore, our scaled form (28) implies that the system matrix is (almost) symmetric, and that the simple conjugate gradient method (CG) is converging as it is applied to a positive definite matrix. The actual convergence rates should be independent of the mesh size h, since we assume that the first order differential operators Th m are inverted exactly. A simple model analysis in [7] shows additionally no dependence on the number M of discrete ordinates. Only the size of the parameter K and the ratio A/K determines the convergence speed of our solution algorithm.

These theoretical statements will be the subject of the following section where some numerical results are presented. In addition, a comparative study concern- ing the computational efficiency of our methods will be carried out in order to identify the method best suited for our purpose.

4. Numerical Results

We consider the 2 + 2-model of radiation from star(s) through dust clouds which is of astrophysical interest (see for instance [8]). The 2 + 2-model is a weaker form of the full 3 + 2-model since the underlying spatial domain ~2 is only two-dimensional. From the mathematical point of view we can expect the same results with much less computational effort, only the physical meaning may be

Mean Intensity Approach for the Numerical Solution of the Radiative Transfer Equation 33

different. In a forthcoming paper, we will present the same numerical test for the full problem the implementation of which is currently in work. Our (physi- cal) model example is the following: In an (artificial) domain 12 c R 2 there are three stars with different radii and different temperatures. Around these stars we first have holes where all material is melt away, then a rapidly decreasing dust cloud appears which largely diminishes the emission intensity. We assume no background radiation which means we have homogeneous boundary values on all F~-. In a preprocessing grey calculation, we compute a temperature T(x) at all points x ~ 12.

To be more precise, our computational domain in Fig. 1 is part of the rectangle [ - 1, 2] • [ - x/3-/2, f3 /21 , and the centers of the stars are (0, 0), (0.5, 0) and (0.75, v~-/4). For a more common astrophysical formulation we rewrite Eq. (1) as

no,. VxI + (K + A)I= A J + Kf, (30) where the (dimensionless) parameters K, A and f are given in Table 1 (with o- = 5.67.10 -5 while the terms r i are a measure for the distance of the i-th star). The c~ are free parameters which will be fixed later. Having calculated the mean intensity J(x), we can determine the corresponding temperature T(x) for this grey calculation by

o-Z4(x) ~-- 7rJ(x). (31)

With this temperature field, we calculate in the next step the corresponding intensitites for certain frequencies v using the Kirchhoff-Planck function By(T) as source term. Bv(T) is of the following form (with h, c, K the usual physical constants, see [4])

2hv3 m, -1

--( I B~(T)= cZ e ~ - 1 (32)

The frequencies v are taken in the interval [1013, 10 is] and the corresponding values for K~, I~ and f~ are taken from Table 2. The reader should note that this approach does not satisfy the equation for temperature equilibrium (see [4]), which is of physical interest. Therefore, some more iterations of calculating first a temperature field by given intensities, and then the corresponding frequency- dependent intensities have to be performed. This more consistent procedure would lead to slightly modified results in Fig. 3. With given values for c~, we are able to change the ratio for ~-~--~ -~-1+c, which essentially determines the convergence speed of our solution methods. We choose c~ E {1, 10, 100, 10000} which should lead (following our previous theoretical considerations) to conver- gence rates P for the usual approximate A-iteration with P = 0.5, 0.9, 0.99, 0.9999. However, this result will only be true if the total extinction K + ~. is sufficiently large which can be controlled by the parameter V/Vma x.

In Fig. 2, we show the convergence rates for the different value of ca. The underlying computational domain is the coarse grid in Fig. 1 with about 2,000

34 S. Turek

Figure l. Computational domain with coarse grid and after two refinements

gridpoints. Our grid is adapted to the size of the parameter functions K and A as the construction process is determined by the local opacity on element level. Since the mesh adapts to the mean intensity and not to the specific intensities, we call it J-adaptiue. This grid generation is still a pure preprocessing step but work is being done to perform the same in a fully adaptive way. My special thanks goes to G. Kanschat and F.T. Suttmeier who are presently developing the grid generator and this fully adaptive version.

Mean Intensity Approach for the Numerical Solution of the Radiative Transfer Equation

Table 1. Parameters for grey calculation

I radius

K

h

f

35

starl

0.01

100

o" 60004

star2

0.03

100

o" 30004

star3 hole1 hole2 [ hole3 cloud

0.01 0.05 0.1 0.05

100 0 0 0 ,c~(llr~ + llr~ + llr~)

0 0 0 0 c~K

0"50004 0 0 0 0

K v

h~

L

Table 2. Parameters for frequency dependent calculation

star1

100

0

B~(6000)

star2 star3

100 100

0 0

B,(3000) B.(5000)

holel h o l e 2 hole3 cloud

0 0 0 v/vm,,x. K(grey)

0 0 0 c a K~

0 0 0 By(T)

For the ordinate discretization, we take the trapezoidal scheme (7) with K - - 8 and L -- 9 which is enough for these mathematical tests. The discrete frequen- cies are N, = 11 logarithmically equidistant points in the interval [1013, 10~s]. The first convergence rate in the diagrams is for the grey calculation and should be similar to the rate for the largest frequency. Our notation reads as follows: lambda corresponds to the approximate A-iteration, rich to the weighted Richardson scheme (mostly o- h is chosen as 1.2); cg and pcg are the simple CG and its preconditioned version by the diagonal matrix, analogously bicg and pbicg.

The results show perfectly the predicted behaviour for the approximate A-itera- tion: For large scattering this method becomes useless ( p _> 0.99). The very simple local preconditioning leads to better results, but in our case when the size of the parameters is changing dramatically over the domain ~2, the results are far from being satisfying. Only the CG-like methods, and here especially the preconditioned Bi-CGSTAB, lead to useful results for all parameters. Since these methods consist of only one or two matrix-vector multiplications in each iteration step plus some scalar products, they require (almos0 the same numeri- cal effort as the classical Richardson schemes with only several more auxiliary vectors (2-4) being needed. This is also confirmed by our numerical tests which show that one CG step costs (on a sequential computer) about 10 percent more than one A-iteration, and one Bi-CGSTAB about 15 percent more than two A-steps.

For the next calculations, which are more of astrophysical interest, we use our

36 S. Turek

a

tristar 1 0.5

0.3

0.2

0.1

b tristar_l 0

0.7 0.6 "rich"

o .a \ , v " - . , , " . . . . . . . . "lbicg', 0.2 N ~ " ~ . _ . . , ~ ..~x,-"" . . . . . . "z"= "plzie9" 0.1

C

1

0.8

0.6

0.4

0.2

tristar_l O0 d tristar 10000

0.8 ,f " '~::cz:. . . ." "rich" / , , / r - - ~ . _ _ _ _ . . . ~ ; . 2 . . ~ . . . . . . . . . "lam"

.. ,,peg. 0.4 h . . . . . . . . . . . . -... "pbicg" 0.2

Figure 2. Convergence rates for the first (mathematical) test

favourite method, the locally preconditioned Bi-CGSTAB iterations. We calcu- late on the refined grid with about 26,000 gridpoints, and 240 discrete ordinates (16 • 15). Hence the number of unknowns for each frequency step is about 6.2.106 , and these calculations are done for 21 frequencies such that the overall number of equations amounts to about 108 . All these calculations are carried out in F O R T R A N 77 on a SUN Sparc 10/41 in less than one day (between 5 and 20 hours), and the amount of storage is about 15 Megabytes in the fastest version.

Before the results are shown, let us make some remarks concerning the stopping criteria since they may be essential for the speed of our algorithm. We use the standard ones: The (relative) residual and the maximum of relative changes must be small enough,

flAhJ~h ~ --FhlL <-- eallFhlL, IIJh ~~ -- J~'-I)IL _< e2[IJ~h~)[L, (33)

with el = e 2 = 10 -8. These stopping criteria may be too weak if we consider additionally the physically interesting value of the observed total intensity (in direction of a given no;)

= I(x,oy)nx-n , dx. (34)

This is the physical value we are most interested in and which is shown in Fig. 3. For very large extinction and scattering it may happen that the residual and relative changes have to be calculated up to a tolerance of almost 10 -3o to ensure the relative changes for Z(w') to be less than 10 -5, while for smaller opacity this limit is achieved even for residuals the size of 10 - : . So, both criteria have to be combined in a clever way. This can be done since in the case of large opacity when finer residuals have to be computed, our convergence rates are much better than in the other case. In our calculation, this was the case for

Mean Intensity Approach for the Numerical Solution of the Radiative Transfer Equation 37

small and large frequencies when the total intensity E(o/) is small. In this case (see Fig. 3), the convergence rates are better than for the intermediate frequen- cies where E(oY) is large, but can be precisely approximated with a few iterations only. With the right combination we get optimal efficiency for all parameters since unnecessary work is avoided.

5. Conclusions

An improved solution technique has been introduced for the solution of the radiative transfer equation. The advantage compared to the more general approach in [7] is that the discrete problems are much smaller. Since formally the dependence of the specific intensities is eliminated, the amount of storage is decreased while the computational expense is about the same. Another (theoret- ical) advantage is that with this approach a convergence result for all parameters

IO TM,

+toI 3,

i o I ~

>~ 1011`

i010 '

10 g

10 8

10 ?

trlstar'3 2~_1 trlstar3 240 100

10t3.

lO 12.

lO 11.

"~ l o l O .

! 1o g

lO 8

, = , , , , i , i , , , , " , , , , , i , ; , 107 I 2 3 4 5 G 7 s 9 1011121314151~;171819202122

f r ~ l U O e C y

1 2 3 4 5 8 7 8 9 I01112131415161710192021;~2

trKluency

1013

1012

1011 �9

:h

" ; 101~

log �9

10 8

107

trlstar3 240 10

0,5,

" 0.4, I

0,3,

~ .

0.1-

1 2 3 4 w 6 7 8 g 101112131415161718192(]2122

trlstar3 240_cony

1 2 3 4 $ 6 7 8 9 10111213141516171819202122

f r K l u e n c y

Figure 3. Total intensities and corresponding convergence rates. The directions are co 1 = ( - 1 , 0, T 2 T - 3 0 ) , ~O = ( 0 , - - 0 . 5 , 0 . 8 6 6 ) a n d o) = ( 0 , - 1 , 0 ) r

38 S. Turek: Numerical Solution of the Radiative Transfer Equation

can be ensured. The meaning for practical applications is that much more complex and hence more realistic problems can be solved, even on computers of workstation type. This approach provides the basis for more complex calcula- tions, for instance with general phase scattering functions R (by defect correc- tion techniques), with an implicit frequency coupling and with a (nonlinear) constraint of radiative equilibrium. These problems and the full 3 + 2-model are in work and will be presented in forthcoming papers.

References

[1] Fiihrer, Chr.: A comparataive study of finite element solvers for hyperbolic problems with applications to radiative transfer. Technical report SFB 359, 65, University Heidelberg, 1993.

[2] Johnson, C., Pitkfiranta, J.: Convergence of a fully discrete scheme for two-dimensional neutron transport. SIAM J. Numer. Anal. 20, 951-966 (1983).

[3] Kalkofen, W.: Numerical radiative transfer, Cambridge: Cambridge University Press 1987. [4] Mihalas, D., Weibel-Mihalas, B.: Foundations of radiation hydrodynamics. Oxford: Oxford

University Press, 1984. [5] Steiner, O.: A rapidly converging temperature correction procedure using operator perturbation.

Astron. Astrophys. 231, 278-288 (1990). [6] Turek, S.: Tools for simulating nonstationary incompressible flow via discretely divergence-free

finite element models. Int. J. Numer. Meth. Fluids. 18, 71-105 (1994). [7] Turek, S.: An efficient solution technique for the radiative transfer equation. Imp. Comput. Sci.

Eng. 5, 201-214 (1993). [8] Turek, S., Wehrse, R.: Spectral appearance of dust enshrouded stars: A combination of a

strongly accelerated solution technique with a finite element approach for 2D radiative transfer, Astron. Astrophys. in press (1994).

[9] Van der Vorst, H.: BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631-644 (1992).

[10] V~ith, H.M.: Three-dimensional radiative transfer on a massively parallel computer. Astron. Astrophys. 284, 319-331 (1994).

Dr. S. Turek Institut fiir Angewandte Mathematik Universit~it Heidelberg Im Neuenheimer Feld 294 D-69120 Heidelberg, Federal Republic of Germany