Commu. Theor. Phys. 59 201 782 788 Vol. 59, No. 6, Jue 15, 201 High-Precisio Direct Method for the Radiative Trasfer Problems ZHANG Ya, 1 HOU Su-Qig ã, 2, YANG Pig, 1 ad WU Kai-Su Ñ 1, 1 Beijig Uiversity of Chemical Techology, Beijig 100029, Chia 2 Istitute of Moder Physics, Chiese Academy of Scieces, Lazhou 70000, Chia Uiversity of Chiese Academy of Scieces, Beijig 100049, Chia Received November 16, 2012; revised mauscript received April 22, 201 Abstract It is the mai aim of this paper to ivestigate the umerical methods of the radiative trasfer equatio. Usig the five-poit formula to approximate the differetial part ad the Simpso formula to substitute for itegral part respectively, a ew high-precisio umerical scheme, which has 4-order local trucatio error, is obtaied. Subsequetly, a umerical example for radiative trasfer equatio is carried out, ad the calculatio results show that the ew umerical scheme is more accurate. PACS umbers: 95.0.Jx, 02.60.Lj Key words: radiative trasfer equatio, direct method, five-poit umerical formula, trucatio error 1 Itroductio The radiative trasfer is a fasciatig problem i uclear astrophysics ad astrophysics. The mathematical modelig of radiative trasfer i which the pheomea of absorptio, emissio ad scatterig are tae ito accout is sometimes made usig the itegral-differetial equatio, also as radiative trasfer equatio RTE. The ey problem of radiative trasfer theory is to solve the RTE, the the physical structure ad chemical compoet of the stellar atmospheres ca be obtaied by solvig the RTE. [12] I past fifty years, may computatioal methods for solvig the RTE problem have bee preseted. I the direct methods [] for the umerical solutio of RTE, which directly approximate the equatio with the umerical differetiatio formula for the differetial part ad the umerical itegratio formula for the itegral part, the RTE is coverted to a group of algebraic equatios which are coveiet to solve. But it is time cosumig to solve the algebraic equatio. So iterative solutio techiques have received more attetio ad are becomig the most popular method for the solutio of the RTE. However, the classical iteratio i.e. so-called Lambda iteratio [4] is still ot a perfect method because of slow covergece. [5] Cao [67] employed the accelerate iterative scheme of radiative trasfer to overcome the difficulty. Later it was geeralized as accelerate λ iteratio [8] ALI. Superior iteratio method of the radiative trasfer based o Gauss- Seidel GS ad successive overre-laxatio SOR were developed by Bueo et al. [910] Actually, iterative method has some limitatios i itself: O the oe had, oe eed to select the suitable iitial value for successful computatio i order to be coverget for iteratio. O the other had, the selectio of accelerate factors is also a difficult problem. [11] I additio, the Mote Carlo calculatio method is also a importat tool for the radiative trasfer equatio, [12] especially i the multi-dimesioal geometries ad scatterig. [1] However, the major drawbac of the covetioal Mote Carlo method is the log computatioal time for coverged solutio. [14] The direct methods is advatageous to solve the RTE. Firstly, it is udemadig to deduce the calculatio formula of RTE. Secodly, the algebraic equatios which come from discretizatio of RTE ca be solved easily by usig the public pacage, such as Matlab, Maple, Fortra ad so o. Ideed, the direct methods, especially the calculatio of the algebraic equatios cosume a great deal of machie time of computer. However, with great developmet of the computer techology, the costrait of computatioal speed ad machie time has bee overcome, so it is ecessary to discuss here the direct method of radiative trasfer equatio oce agai. Dig [15] et al. itroduced a ew computatio scheme of direct method for RTE. I their research, the derivative part ad itegral term are respectively used to be substituted for three poit cetral differece scheme ad complicated Simpso formula. But we ca otice that though the complicated Simpso formula has a 4-order precisio ad the three poit cetral differece has a - order local error, the fial precisio of the umerical solutio ca t reach 4-order. I fact, the calculatio error is cotrolled by the miimum-order of the discrete process. So if the error-order for the umerical differetiatio ad Supported by the Youth Foudatio of Beijig Uiversity of Chemical Techology uder Grat No. QN0622 E-mail: wus@mail.buct.edu.c c 201 Chiese Physical Society ad IOP Publishig Ltd http://www.iop.org/ej/joural/ctp http://ctp.itp.ac.c
No. 6 Commuicatios i Theoretical Physics 78 the umerical itegratio ca eep cosistet, the optimal algorithm could be obtaied. I the preset paper we developed a ew scheme, which uses a five-poit umerical formula ad complicated Simpso formula to replace the differetial part ad itegral term respectively. The reaso ew method is superior to other method is that a five-poit umerical formula has a 4-order precisio, which is twice as accurate as three poit cetral differece scheme. So usig this scheme is just matchig the precisio of complicated Simpso method, the fial precisio of the method ca achieve 4-order. The model of RTE ad the calculatio formula is itroduced i Sec. 2. I Sec., a umerical example for RTE is computed, ad the calculatio results show that our ew umerical scheme is more effective. A coclusio i preseted i Sec. 4. 2 The Model of RTE ad the Calculatio Method The model cosidered here is a coheret aisotropic scattig term i plae-parallel slab. Uder this coditio, the radiative trasfer equatio ca be writte as the followig form, [12,1516] µ Iη, µ = Iη, µ + c 2 1 1 Pµ, µ Iη, µ dµ, 1 where fuctio I deotes the desity of the radiative particles, η is the distace ormalized by the optical depth of of the layer, [1516] ad c is the albedo for siglescatterig, ad µ deotes the directio cosie of the agle made by the specific itesity at ay depth τ with the directio of icreasig τ, Pµ, µ is the scattig phase fuctio respectively. [12,1516] The boudary coditios of Eq. 1 are Iη, µ = φ 1 µ, at η = 0 0 < µ 1, Iη, µ = φ 2 µ, at η = 1 1 µ < 0. 2 We ow covert the differetial-itegral equatio ito a set of equatios by discretizatio of all variables. Here, is positive eve. We divide [0, 1] ad [1, 1] ito ad 2 subitervals of equal legths h = 1/, respectively, amely η i = ih, i = 0, 1, 2,..., ; µ j = jh, ν = h, j, = 0, ±1, ±2,..., ±. For ay variable g, we write g z d, µ m, ν = g dm, amely η i, µ j = i, j, Iη i, µ j = I ij, Iη i, ν = I i, Pµ j, ν = P j. At every ode i, j, Eq. 1 ca be rewritte as µ j I = I ij + c 1 Pµ j, νiη i, νdν. i,j 2 Approximatig the derivative with five-poit umerical differetial formula, 1 I = 1 i,j 12h I i2,j 8I i1,j + 8I i+1,j I i+2,j, i = 2,,..., 2 ; j = 0, ±1,..., ±, I 0,j 12h 25I 0,j + 48I 1,j 6I 2,j + 16I,j I 4,j, j = 1, 2,...,, I 1,j 12h I 0,j 10I 1,j + 18I 2,j 6I,j + I 4,j, j = 0, ±1,..., ±, I 1,j 12h I 4,j + 6I,j 18I 2,j + 10I 1,j + I,j, j = 0, ±1,..., ±, I,j 12h I 4,j 16I,j + 6I 2,j 48I 1,j + 25I,j, j = 1, 2,...,. 4 Approximatig the itegral with Simpso formula, 1 1 Pµ j, νiη i, νdν = h P ji i + P j I i + 2h P j2m I i2m + 4h P j,2m+1 I i,2m+1. 5 Substitutig Eqs. 4 ad 5 ito Eq., ad usig boudary coditios 2, we ca get the followig results. i Whe i = 2,,..., 2, as j =, I i2, + 2 I i1, + 1 ch 6 P, I i, ch 2ch P,2m I i,2m P,2m+1 I i,2m+1 ch 6 P,I i, 2 I i+1, + I i+2, = 0 ;
784 Commuicatios i Theoretical Physics Vol. 59 as j =, I i2, 2 I i1, ch 6 P,I i, ch 2ch as j = 2 + 1, = /2, /2,..., /2 1, P,2m I i,2m P,2m+1 I i,2m+1 + 1 ch 6 P, I i, + 2 I i+1, I i+2, = 0 ; 2 + 1 22 + 1 I i2,2+1 I i1,2+1 ch 6 P 2+1,I i, ch 2ch 2ch 1 as j = 2, = 0, ±1, ±2,..., ±/2 1, m=/2+1 P 2+1,2m+1 I i,2m+1 + 1 2ch P 2+1,2+1 I i,2+1 ch P 2+1,2m+1 I i,2m+1 ch 6 P 2+1,I i, + I i2,2 4 I i1,2 ch 6 6 P 2,I i, ch 2ch 2ch 1 m= ii Whe i = 0, as j =, 1 + 25 ch 6 P, I 0, ch 4 I, + I 4, = ch as j = 2 + 1, = 1, 2,..., /2, ch 6 P 2+1,I 0, ch P 2+1,2m I i,2m P 2+1,2m I i,2m 22 + 1 I i+1,2+1 2 + 1 I i+2,2+1 = 0 ; 1 P 2,2m+1 I i,2m+1 + 1 ch P 2,2 I i,2 ch P 2,2m+1 I i,2m+1 ch 6 P 2,I i, + 4 I i+1,2 1 P,2m I 0,2m 2ch P,2m ϕ 1 2m + 2ch P 2+1,2m I 0,2m 2ch 252 + 1 + 1 2ch P 2+1,2+1 I 0,2+1 ch 2ch 1 2 + 1 I 4,2+1 = ch P 2+1,2m+1 I 0,2m+1 + as j = 2, = 1, 2,..., /2 1, ch 6 P 2,I 0, ch 1 1 1 P 2,2m I i,2m P 2,2m I i,2m 6 I i+2,2 = 0. P,2m+1 I 0,2m+1 4 I 1, + I 2, P,2m+1 ϕ 1 2m + 1 + ch 6 P,ϕ 1 ; 1 P 2+1,2m+1 I 0,2m+1 P 2+1,2m I 0,2m 42 + 1 2 + 1 I 1,2+1 I 2,2+1 + P 2+1,2m ϕ 1 2m + 2ch P 2,2m I 0,2m 2ch 1 42 + 1 I,2+1 P 2+1,2m+1 ϕ 1 2m + 1 + ch 6 P 2+1,ϕ 1 ; P 2,2m+1 I 0,2m+1 + 1 252 ch P 2,2 I 0,2
No. 6 Commuicatios i Theoretical Physics 785 ch 1 2 I 4,2 = ch P 2,2m I 0,2m 2ch iii Whe i = 1, as j =, 1 m= P 2,2m φ 1 2m + 2ch P 2,2m+1 I 0,2m+1 + 8 I 1,2 6 I 2,2 + 8 I,2 I 0, + 1 + 5 ch 6 6 P, I 1, ch P 2,2m+1 φ 1 2m + 1 + ch 6 P 2,φ 1. P,2m I 1,2m as j =, 2ch P,2m+1 I 1,2m+1 ch 6 P,I 1, I 2, + I, I 4, = 0 ; 2 2 ch 6 P,I 1, ch P,2m I 1,2m 2ch P,2m+1 I 1,2m+1 + 1 5 ch 6 6 P, I 1, + I 2, I, + I 4, = ϕ 1 ; 2 2 as j = 2 + 1, = /2, /2 + 1,..., /2 1, 2 + 1 I 0,2+1 ch 6 P 2+1,I 1, ch m=/2+1 52 + 1 + 1 2ch 6 P 2+1,2+1 I 1,2+1 ch ch 6 P 2+1,I 1, + as j = 2, = 0, ±1, ±2,..., ±/2 1, P 2+1,2m I 1,2m 2ch 1 P 2+1,2m I 1,2m 2ch 2 + 1 2 I 2,2+1 2 + 1 2 I,2+1 + 2 + 1 I 4,2+1 = 0 ; 2 I 0,2 ch 6 P 2,I 1, ch + 1 5 ch P 2,2 I 1,2 ch 1 P 2,2m I 1,2m 2ch P 2,2m I 1,2m 2ch ch 6 P 2,I 1, + I 2,2 I,2 + 6 I 4,2 = 0. iv Whe i = 1, as j =, as j =, I 4, I, + I 2, + 2 2 ch P,2m I 1,2m 2ch 1 m= 1 5 ch 6 6 P, I 1, I 4, + I, I 2, ch 2 2 6 P,I 1, ch 2ch P 2+1,2m+1 I 1,2m+1 P 2+1,2m+1 I 1,2m+1 P 2,2m+1 I 1,2m+1 P 2,2m+1 I 1,2m+1 P,2m+1 I 1,2m+1 ch 6 P,I 1, = ϕ 2 ; P,2m+1 I 1,2m+1 + 1 + 5 6 ch 6 P, I 1, + I, = 0 ; P,2m I 1,2m
786 Commuicatios i Theoretical Physics Vol. 59 as j = 2 + 1, = /2, /2 + 1,..., /2 1, 2 + 1 I 4,2+1 + 2 + 1 2 I,2+1 ch m=/2+1 P 2+1,2m I 1,2m 2ch 2 + 1 I 2,2+1 ch 2 6 P 2+1,I 1, 1 52 + 1 + 1 + 2ch 6 P 2+1,2+1 I 1,2+1 ch 2ch as j = 2, = 0, ±1, ±2,..., ±/2 1, P 2+1,2m+1 I 1,2m+1 P 2+1,2m I 1,2m P 2+1,2m+1 I 1,2m+1 ch 6 P 2+1,I 1, + 2 + 1 I,2+1 = 0 ; I 4,2 + I,2 I 2,2 ch 6 6 P 2,I 1, ch 2ch 2ch 1 m= v Whe i =, as j =, 1 P 2,2m+1 I 1,2m+1 + 1 + 5 ch P 2,2 I 1,2 ch P 2,2m+1 I 1,2m+1 ch 6 P 2,I 1, + 2 I,2 = 0. I 4, 4 I, + I 2, 4 I 1, ch 2ch = ch 0 as j = 2 + 1, = 0, 1, 2,..., /2 1, P,2m+1 I,2m+1 + 1 + 25 P,2m ϕ 2 2m + 2ch 1 ch 6 P, 2 + 1 42 + 1 2 + 1 I 4,2+1 I,2+1 + I 2,2+1 ch ch = ch 0 as j = 2, = 1, 2,...,/2 1, P 2+1,2m I,2m 2ch P 2+1,2m I,2m 2ch 1 P 2+1,2m φ 2 2m + 2ch P,2m I,2m I, P 2,2m I 1,2m P 2,2m I 1,2m P,2m+1 ϕ 2 2m + 1 + ch 6 P,ϕ 2 ; 42 + 1 I 1,2+1 252 + 1 P 2+1,2m+1 I,2m+1 + 1 + 2ch P 2+1,2+1 I,2+1 P 2+1,2m+1 I,2m+1 ch 6 P 2+1,I, 1 I 4,2 8 I,2 + 6 I 2,2 8 I 1,2 ch 2 2ch 1 P 2+1,2m+1 φ 2 2m + 1 + ch 6 P 2+1,φ 2. 1 P 2,2m+1 I,2m+1 + 1 + 25 ch 6 P 2,2 I,2 ch P 2,2m I,2m P 2,2m I,2m
No. 6 Commuicatios i Theoretical Physics 787 2ch + 2ch m= 1 P 2,2m+1 I,2m+1 ch 6 P 2,I, = ch 0 P 2,2m+1 φ 2 2m + 1 + ch 6 P 2,φ 2. P 2,2m φ 2 2m The system of liear algebraic equatios after discrete are 2 2 + 1 order ad the uow umber are I 0,,..., I 0,1 ; I 1,,...,I 1,0,...I 1, ;...;I 1,,..., I 1,0,... I 1, ; I,1,...,I,. The, we ca solve the liear algebraic equatios i i v with the public pacage, such as Matlab, Maple, Fortra ad so o. [17] Numerical Example As a applicatio of our umerical calculatio method, we compute the umerical solutio for a radiative trasfer equatio. For the aisotropic scatterig radiative trasfer, customarily, the scattig phase fuctio i Eq. 1 ca be tae the followig form, [2,18] Pµ, µ = 1 + a m P m µp m µ, 6 where P m is the m-degree Legedre polyomial, ad the a m are expasio coefficiets. Here, cosistetly with Refs. [15 16], we tae c = 1, scattig phase fuctio are defied as F + η = 2 1 0 Iη, µµdµ. 7 I Table 1, the radiative fluxes F + 1 value at the lower boudary have bee give for differet optical thicess divided by differet ode umber = 4, 6, 8. The results are compared with the exact value [16] listed i last lie of Table 1. Table 2 presets the radiative fluxes at the lower boudary for differet optical thicess for aisotropic coefficiet a 1 = 0.7, 0.0, 0.7, here = 8. Meawhile, exact value [16] ad the umerical solutio from referece [15] are also listed i order to cotrast with our ew method. It is evidet from Table 2 that preset method is effective to obtai more accurate umerical solutio tha Ref. [15]. Pµ, µ = 1 + a 1 P 1 µp 1 µ ad boudary coditios ϕ 1 µ = 1, µ > 0; ϕ 2 µ = 0, µ < 0. Usig our umerical method, we perform the calculatio of radiative fluxes F + 1 at the lower boudary of the slab for differet value of optical depths ad liear aisotropic phase fuctio. Here, radiative fluxes 4 Coclusio The RTE plays a ey role i the theory of radiative trasfer, so how to solve the RTE efficietly is oe of the most importat problems of the theory of radiative trasfer. With five-poit formula to substitute the differetial part ad the Simpso formula to calculate the itegral part respectively, we get a ew umerical method for the RTE. Our computatio formulae have 4-order local trucatio error. The, we give a umerical example for RTE. By the compariso of umerical results with the previous results, oe ca see that the solutio of the preset method more approach the exact solutio.
788 Commuicatios i Theoretical Physics Vol. 59 I additio, the advatage of the preset method is that the formula is easy to derived, code is ot complicated ad we mae use the public program pacage too. Refereces [1] R.Q. Huag, Theory of Stellar Atmosphere, Yua Peoples Press, Kumig 1986 i Chiese. [2] S. Chadrasehar, Radiative Trasfer, Dover Publicatios Ic, New Yor 1960. [] J.V. Dave, J. Atmos. Sci. 2 1975 790. [4] R.J. Rutte, Radiative Trasfer i Stellar Atmospheres, 8th ed. Utrecht Uiversity Lecture Notes, Utrecht 200 p. 122. [5] D. Mihalas, Stellar Atmospheres, 2d, Freema ad Compay, Sa Fracisco 1978. [6] C.J. Cao, Astrophys. J. 185 197 621. [7] C.J. Cao, J. Quat. Spectrosc. Radiat. Trasfer. 1 197 627. [8] G.L. Olso, L.H. Auer, ad J.R. Buchler, J. Quat. Spectrosc. Radiat. Trasfer. 5 1986 41. [9] J.B. Trujillo ad B.P. Fabiai, Astrophys. J. 455 1995 646. [10] M. Sampoora ad J.B. Trujillo, Astrophys. J. 712 2010 11. [11] I. Hubey, i Stellar Atmosphere Modelig, ASP Coferece Series, eds. I. Hubey, D. Mihalas, ad K. Werer, Astroomical Society of the Pacific, Sa Fracisco 288 200 17. [12] S. Audic ad H. Frisch, J. Quat. Spectrosc. Radiat. Trasfer. 50 199 127. [1] B. Šurla, W.R. Hama, J. Kubát, L.M. Osiova, ad A. Feldmeier, Astro. Astrophys. 541 2012 A7. [14] A. Kersch, W. Morooff, ad A. Schuster, Trasport Theor. Stat. Phys. 2 1994 1001. [15] P.Z. Dig ad Y.K. Mu, Chiese J. Comput. Phys. 9 1992 817. [16] A. Daya ad C.L. Tie, J. Heat Trasfer. 97 1975 91. [17] I. Chivers ad J. Sleightholme, Itroductio to Programmig with Fortra, Spriger Press, Lodo 2008. [18] M.P. Zorzao, A.M. Macho, ad L. Vazquez, Appl. Math. Comp. 164 2005 26.