ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION

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1 ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION HAO GAO AND HONGKAI ZHAO Abstract. We analyze a nuerical algorith for solving radiative transport equation with vacuu or reflection boundary condition that was proposed in [4] with angular discretization by finite eleent ethod and spatial discretization by discontinuous Galerkin or finite difference ethod.. 1. Introduction Radiative transport equation (RTE) (1.1) odels a wide range of applications, such as neutron transport, atospheric radiative transfer, heat transfer and photon igration in tissues. In this paper, we study a nuerical algorith for the following RTE on a bounded doain Ω with sooth boundary Ω (1.1) ŝ Φ( x, ŝ) + µ t Φ( x, ŝ) = µ s f(ŝ, ŝ )Φ( x, ŝ )dŝ + q( x, ŝ), ( x, ŝ) Ω S, where Φ( x, ŝ) is the particle density at position x along direction ŝ, q is the source ter, S denotes the angular space, i.e., unit circle in two diensions (D) S 1 or unit sphere in three diensions (3D) S. The paraeters in (1.1) are absorption coefficient µ a, scattering coefficient µ s, transport coefficient µ t = µ a + µ s and scattering kernel f. We assue all the paraeters in (1.1) are non-negative and bounded. In particular, µ a is bounded below by a positive constant C µ, i.e., 0 < C µ µ a. Very often f is rotationally invariant, i.e., f(ŝ, ŝ ) = f(ŝ ŝ ), and is noralized with f(ŝ ŝ )dŝ = 1. Although we assue this rotational invariance in this paper for siplicity, our algorith and analysis can be extended for the general case. As an exaple, the coonly used f in optical iaging to odel photon transport in tissues is the following Henyey-Greenstein (H-G) function (1.) f(ŝ ŝ ) = { 1g π(1+g gŝ ŝ ), n = 1g 4π(1+g gŝ ŝ ) 3/, n = 3, where the paraeter 0 g 1 indicates how forward peaking the scattering is. To specify the boundary condition, we define ˆn( x) as the boundary noral for x Ω, Γ + (Γ ) = {( x, ŝ) Ω S, with ŝ ˆn > 0( 0)} as the outgoing (incoing) Key words and phrases. Radiative transfer equation, discrete ordinate ethod, source iteration, Henyey-Greenstein function, discontinuous Galerkin ethod. The work was partially supported by NIH/NIBIB grant EB and NSF grant DMS

2 HAO GAO AND HONGKAI ZHAO boundary, and n i (n o ) as the refraction index of the ediu (environent). Then the reflection boundary condition for (1.1) is (1.3) Φ( x, ŝ) = R( x, ŝ, ŝ )Φ( x, ŝ ), ( x, ŝ) Γ, ( x, ŝ ) Γ +, where ŝ refers to angular direction at x Ω that is reflected into ŝ at x when n i n o, and R corresponds to the reflection energy ratio coputed through Fresnel forula [5]. When there is no refraction index isatch, i.e., n i = n o, (1.3) is the so called vacuu boundary condition with R = 0. With the aforeentioned conditions on µ a, µ s and f, there exists a unique positive solution to RTE (1.1), when Q( x, ŝ) = d 0 q( x rŝ, ŝ)e r 0 µt( xr ŝ)dr dr is bounded and positive [6]. Here d is the distance fro x to the doain boundary Ω along the direction ŝ.. Nuerical Algoriths The analytical solution for RTE is rarely available except in very special setup. Efficient nuerical algoriths are in need for any applications. However, the efficient nuerical coputation of RTE is challenging ainly due to the following difficulties: high diensionality; different solution behavior, e.g., fro the transport regie near the source to the diffusion regie after a few ean free paths; forward peaking due to anisotropic scattering kernel function; general boundary conditions. Moreover, a large and not so sparse (due to the scattering ter) linear syste has to be solved after discretization that requires an efficient atrix solver. In [4], we developed an efficient ultigrid nuerical algorith for RTE that works for general geoetry and general reflection boundary condition. The solver is available at The key features of our ethod include angular and spatial discretization that (1) preserves the correct behavior of transport and scattering operator, () captures both short and long range interactions in spatial and angular doain effectively when using the ultigrid strategy; iproved source iteration (ISI) based on Gauss-Seidel iterations with angledependent ordering of spatial nodes which (1) captures transport efficiently, () converges faster than source iteration (SI); ultigrid ethod in both angular and spatial doain with ISI as the relaxation which further accelerates the solution algorith, particularly in the diffusion regie; flexibility for both structured and unstructured grids in D and 3D with general boundary conditions. In the following, we will briefly describe our solution algoriths for RTE and the details can be found in [4]..1. Angular discretization. Let {L (ŝ), 1 M} be the piecewise linear angular basis on an angular esh {ŝ, 1 M} of S, and w = L dŝ.

3 ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION 3 Then we approxiate the scattering angular integration I = f(ŝ ŝ )Φ(ŝ )dŝ by (.1) Ĩ = f Φdŝ, where f and Φ are the linear approxiation of f and Φ respectively. We start with angular discretization (Figure 1) for D spatial doain, where S 1 is equally divided into angular intervals {θ, 1 M} with interval length h a, and therefore is also rotationally invariant in ters of h a. The angular discretization of the scattering suation by the finite eleent ethod (FEM) goes as follow. For each ŝ, Ĩ = f(θ θ ) Φ(θ )dθ. Moreover, since Φ = Φ L, we have Ĩ = w0 Φ with w0 = f(θ θ )L dθ. Then we noralize the weight w 0 s to get w s, i.e., w = w0 /( w0 ) so that w = 1. In suary, the angular scattering suation is discretized as M (.) I ha = w Φ, 1 M. =1 Please note that w = w due to the rotational invariance of f and the unifority of the angular esh. Direction M Direction Direction M + 1 Direction 1 Direction M Figure 1. Angular discretization in D In 3D, the angular esh on the unit sphere S is the projection of eight triangular planes, each of which belongs to a different quadrant. The esh on the triangular plane in the first quadrant is shown in Figure. And then the esh is projected onto the unit sphere which gives a polyhedral approxiation of the sphere. This siple triangulation is quite unifor although it is not rotationally invariant. Moreover it is easy to coarsen and refine for ultigrid ethod in angular doain. The discretization is siilar to D and the weight w 0 is coputed on

4 4 HAO GAO AND HONGKAI ZHAO the polydedron with the piecewise linear approxiation f defined on each eleent of the polyhedron. Then the weight w is noralized fro w 0 as in D. Note that w = w is not the case in 3D due to the violation of the rotation invariance of the angular esh in 3D. Coparing with the angular discretization through spherical haronic expansion, the direct use of an angular esh allows to (1) capture both short and long range interactions effectively when using the ultigrid ethod, () deal with general reflection boundary condition at ease, which consequently is transfored into soe local dependence in angular doain, e.g. through Snell s law. Moreover, piecewise linear approxiation guarantees that w 0, which preserves the qualitative behavior of localized scattering even if the angular esh resolution ay not be enough in an extreely forward peaking situation. We will estiate the error between φ after the angular discretization (.3) and RTE solution Φ in section 3.1. Reark.1. Although an octahedron with eight equilateral triangular faces are used for angular discretization, for better unifority, an icosahedron, a regular polyhedron with 0 identical equilateral triangular faces, can be used. z 1 x o y Figure. Angular discretization in 3D

5 ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION 5.. Spatial discretization. Let {φ ( x), 1 M} be the RTE solution along the angular grid {θ, 1 M} after angular discretization. (1.1) is now the following coupled syste of first-order partial differential equations in space M (.3) ŝ φ + µ t φ = µ s w φ + q, 1 M, =1 where q is the source ter along the direction ŝ, i.e., q = q( x, ŝ ), and the scattering integral is approxiated by the suation with the scattering weight {w, 1, M}. Notice that w s are all positive since they are coputed through piecewise linear FEM for non-negative functions. Next we discretize the syste (.3) in space. In [4], we chose the Discontinuous Galerkin (DG) ethod for the unstructured grid (see section 3. for the DG discretization). DG was initially developed for scalar transport equation [8] for neutron transport, and it deals with hyperbolic probles such as transport equation effectively due to its natural upwind nature as well as flexibility for high-order approxiation and esh geoetry. Besides, in ipleentation there is no need to for global atrix explicitly for DG and the solution is solved by successively sweeping through individual esh eleents. We will analyze DG discretization of the syste (.3) on unstructured esh in section 3.. On the other hand, we chose the finite difference ethod (FDM) for the structured grid [4]. For exaple, with the first-order upwind FDM on the D spatial doain when θ [0, π ), we have M (.4) (a + b + µ t )φ i,j, (aφ i1,j, + bφ i,j1, ) µ s w φ i,j, = q i,j, cos θ x sin θ y =1 where a = 0, b = 0, and i, j are indices for x and y coordinate respectively.we will analyze FDM for the syste (.3) on structured esh in section Iproved source iteration. SI is a popular ethod for solving the discretized linear syste of RTE. In SI, the scattering entirely lags one step behind transport, i.e., (.5) (ŝ + µ t )Φ n+1 ( x, ŝ) = µ s f(ŝ, ŝ )Φ n ( x, ŝ )dŝ + q( x, ŝ). This siple iterative procedure has a clear physical interpretation, e.g., Φ n captures all particles that have been scattered no ore than n ties. However SI has a slow convergence when the scattering is doinant, e.g., in the optical thick regie where the doain size is large in ters of ean free paths. ISI was developed in [4]. ISI is essentially a Gauss-Seidel ethod in both angular and spatial variables, which utilizes the ost updated inforation during iterations. We also use angle-dependent ordering in ISI, i.e., the ordering that aligns with ŝ to follow the characteristic direction of the transport operator. For siplicity, we illustrate SI and ISI using the first-order upwind FDM (.4). Here since θ [0, π ), the ordering in ISI is i = 1,,, j = 1,,. Then ISI is (.6) φ n i,j, = aφn i1,j, + bφn i,j1, + µ s ij ( 1 =1 w φn i,j, + M =+1 w φn1 i,j, ) + q i,j,, a + b + µ t µ s w

6 6 HAO GAO AND HONGKAI ZHAO while the SI is (.7) φ n i,j, = aφn i1,j, + bφn i,j1, + µ s ij M =1 w φn1 i,j, + q i,j, a + b + µ t. It can be easily shown that upwind spatial discretization cobined with angular discretization for general boundary condition (.8) produces a linear syste with M-atrix, which guarantees that (1) ISI converges and converges faster than SI, () global error in axiu nor is under the control, (3) ISI is a soothing operation and with the correct ordering it efficiently handles the solution in transport regie. In section 3.3, we will analyze this FDM schee (.4) and copare ISI (.6) with SI (.7)..4. Boundary condition. Our direct FEM-based angular discretization deals with general boundary conditions at ease. For exaple, if there is a isatch of refraction index at the boundary, local esh-based photon flux in one direction can be reflected and refracted into other directions according to Fresnel forula [5]. To discretize (1.3) at a boundary point x along an incoing direction ŝ, which belongs to the angular esh, there is an outgoing direction ŝ such that φ( x, ŝ ) = R( x, ŝ, ŝ )Φ( x, ŝ ), where ( x, ŝ ) Γ, ( x, ŝ ) Γ +. However, ŝ ay not belong to the angular esh, so we use linear interpolation of fluxes at neighboring angular esh points ŝ to approxiate the flux in direction ŝ. So after angular discretization the reflection boundary condition becoes (.8) φ ( x) = r ( x)φ ( x), ( x, ŝ ) Γ, ( x, ŝ ) Γ +. Two neighboring directions whose interval encloses ŝ in D and three neighboring directions whose triangle encloses ŝ in 3D are used for the linear interpolation. In particular r = R( x, ŝ, ŝ ) 1. When the incoing direction ŝ is alost tangential to the boundary at x, the corresponding outgoing direction ŝ is also alost tangential to the boundary. Thus soe ŝ used in the interpolation ay be a incoing direction. See Figure 3 for a D exaple: ŝ 1 is the reflection angle for ŝ 1, which is not on the angular grid and is interpolated fro ŝ 1 and ŝ. In D it is easy to show that the general for (.8) is still valid. That is φ ( x) = r[αφ + (1 α)φ ( x)], where r = R( x, ŝ, ŝ ) 1 and the weight α > 0 in the linear interpolation. So r(1 α) (.9) φ ( x) = 1 rα φ rφ. In 3D the situation is ore coplicated since three neighboring directions are involved in the linear interpolation. Incoing directions that are alost tangential to the boundary ay not be explicitly represented in ters of outgoing directions, e.g., when two of the three directions used in the interpolation are incoing directions. In suary, the general reflection boundary condition (.8) can be regarded just as soe kind of scattering relation aong different directions at boundary. So our angular discretization plus linear interpolation for general boundary conditions inherits those nice properties for approxiating the scattering ter in RTE, i.e., a true averaging operation aong directions. When coupled with finite difference schee for spatial discretization, it can be easily shown that the fully discretized linear syste has a M-atrix even with general reflection boundary condition (see the proof for Theore 6).

7 ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION 7 ŝ ŝ tissue ŝ 1 air ˆn ŝ ŝ ŝ 1 Figure 3. Reflection at the boundary in D: only the solid lines are on the angular grid. 3. Error Estiates and Convergence Analysis 3.1. Angular discretization. The ain result in this section is to show that the solution φ after angular discretization of (.3) and RTE solution Φ to (1.1) satisfies (3.1) φ Φ = M w [φ ( x) Φ( x, ŝ )] = O(h a) =1 Fro now on we define the following weighted L nor (3.) φ := Ω M w =1 φ Ω In order to prove Theore 1-3, we will first prove the following estiate in D (Lea 1) and 3D (Lea ) (3.3) I ha I = O(h a). Here is with respect to the angular variable. To siplify the notation, h is used to represent the angular esh size h a in this section fro now on. Lea 1. Assuing f, Φ C (S 1 ), the aforeentioned D angular discretization with I h (ŝ ) = w Φ has the second order convergence, i.e., I(ŝ ) I h (ŝ ) = O(h ), for all. Proof. I I h I Ĩ + Ĩ Ih, where Ĩ is defined by (.1). We estiate these two ters separately.

8 8 HAO GAO AND HONGKAI ZHAO First, I Ĩ = S 1 (f f)φ + f(φ Φ)dθ π( f f Φ + f Φ Φ ). Due to the piecewise linear approxiation, f f = O(h ), Φ Φ = O(h ), thus I Ĩ = O(h ). Next, observe that 1 w 0 = (f f)dθ = O(h ) and w 0 S 1 w has the sae sign for all (because w 0 s are all non-negative and w is the noralized w 0, i.e.,w = w0 /( w0 )), Ĩ Ih = (w 0 w )Φ Φ w 0 1 = O(h ). We have proved (3.3) in D. In 3D, the angular discretization (Figure ) is siilar except that the weight w 0 is coputed on the flat eleent of the triangulation of the sphere with piecewise linear approxiation. Then the weight w is noralized fro w 0. Lea. Assuing f, Φ C (S ), the 3D angular discretization as described above with I h (ŝ ) = w Φ has the second order convergence, i.e., I(ŝ ) I h (ŝ ) = O(h ), for all. Proof. Consider an triangle eleent of the triangulation of S as depicted in Figure. Let x be a point on the triangle denoted by and let x r = x/ x be the corresponding projection onto the sphere. The ap is one-to-one, onto and sooth on each triangular eleent. In particular, x r x = x r (1 x ) with 1 x = O(h ). The change of variable x r ( x) is sooth and x x r = 1 x (I x x x x ) It can be easily seen that I x x x x is alost the projection to the triangle plane up to a O(h) perturbation. So the corresponding surface area on the sphere and the surface area on each triangle eleent satisfies ds( x r ) = ds( x)(1 + O(h )). Also the piecewise-linear approxiation f( x) of any function f( x r ( x)) C ( ) on each triangle satisfies f f = O(h ). Therefore, I = fφ ds( x r ) = [ f + O(h )][ Φ + O(h )][1 + O(h )] ds( x) = Ĩ + O(h ), S i i where Ĩ = i i arguent as in Lea 1, Ĩ(ŝ ) I h (ŝ ) = O(h ). f Φds( x). In particular Ĩ(ŝ ) = w0 Φ. Use the sae Reark 3.1. H-G function (1.) belongs to C for g < 1. When g = 1, f is a Delta function in angle. So there is no scattering aong different directions and I(ŝ) = Φ(ŝ). As a result, the syste (.3) is reduced to a set of uncoupled firstorder transport equations, which can be solved easily and the analysis is the sae as that for a single transport equation. Next we will use Lea 1 and to prove the second-order convergence (3.3) for the angular discretization in Theore 1. However, due to no rotation invariance for the triangulation on the angular sphere, we need the following lea in order to

9 ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION 9 proceed in 3D. For coparison, the grid is unifor and rotationally invariant on the circle in D. As a result, w s are equal in D, while they are not in 3D. For the sae reason, the angular weight w s are syetric in D, i.e., w = w which is not true in 3D either. However, this broken syetry in 3D can be reedied in the proof of Theore 3 using the following lea between w and w. Notice that although the lea is for the noralized weight w s, it also holds for w 0 s since w has the sae dilation fro w0 and w = 1 + O(h ) for w 0 all. Lea 3. w w w w = O(h) for angular discretization in 3D. Proof. We have w S 0 = f(ŝ ŝ)l (ŝ)dŝ= ( f(ŝ ŝ )+O(h))L (ŝ)dŝ = ( f(ŝ ŝ )+O(h))w. S Siilarly, w 0 = ( f(ŝ ŝ ) + O(h))w. Therefore, w0 w w0 w = O(h). Fro the fact w w 0 = 1 + O(h ) for each, the clai follows. Theore 1. With the angular discretization (.3), the approxiate solution φ satisfies (3.4) ax x, φ ( x) Φ( x, ŝ ) = O(h ). if the solution Φ( x, ŝ) to the RTE is C (S). Proof. For the siplicity of notation, let Φ denote Φ( x, ŝ ). (3.5) M =1 w φ f(ŝ, ŝ )Φ(ŝ )dŝ = M w (φ Φ ) + D =1 with (3.6) D := w Φ f(ŝ, ŝ )Φ(ŝ )dŝ. Let ε = φ Φ, and then ε satisfies M (3.7) ŝ ε + µ t ε = µ s w ε + D. =1 Since Ω is a bounded closed doain, assue that ax x, φ ( x) Φ( x, ŝ ) = e is obtained for for soe at soe point x. (1) If x is in the interior of Ω, ε ( x) = 0 and because w 0, w = 1, fro (3.7) we have µ t e µ s w e + D µ a e D e = O(h ). () If x belongs the boundary Ω, (a) If ( x, ŝ ) Γ +, i.e., ŝ ˆn > 0, we have ŝ ε ( x) 0, Again fro (3.7) we have e = O(h ).

10 10 HAO GAO AND HONGKAI ZHAO (b) Else if ( x, ŝ ) Γ, for the general reflection boundary condition with discretization (.8) we have ε ( x) = r ε ( x) + D, ( x, ŝ ) Γ, ( x, ŝ ) Γ +, where D is the truncation error due to linear interpolation of the real reflection condition for at the boundary and is of order O(h ). Also r 0 and r 1. Cobined with the arguent for (a) we have e = O(h ). Hence we coplete the proof for (3.4). Theore (Vacuu Boundary Condition). Assue the solution Φ( x, ŝ) to the RTE is C (S). With vacuu boundary condition and sufficiently fine angular esh (3.8) φ Φ = M w [φ ( x) Φ( x, ŝ )] d x = O(h ). =1 Ω Proof. Multiplying both sides of (3.7) with w ε, integrating over the spatial doain and suing over all angular directions, we have 1 M =1 w Ω ŝ ˆnε + M =1 w Ω µ tε (3.9) = M M =1 =1 Ω µ sw w ε ε + M =1 Ω w D ε. With vacuu boundary condition, ε = 0 for ŝ ˆn < 0, so the boundary integral becoes 1 M (3.10) w ŝ ˆnε = 1 w ŝ ˆnε 0. Ω Ω =1 :ŝ ˆn 0 Next we analyze the first ter on the right hand side of (3.9), M (3.11) µ s w w ε ε 1 M µ s w w (ε Ω + ε ) Ω, =1, =1 On one hand,, w w ε = w ε since w = 1 fro our angular noralization; on the other hand, using Lea 3, w w ε = [(w w +w w O(h))]ε = w (1+O(h))ε., Therefore, when h is sall enough, we have, w w ε (1+δ) w ε with δ O(h). Notice that δ = 0 in D due to rotational invariance. Therefore, fro (3.9) and (3.11) we have M M (3.1) w (µ a δµ s )ε w D ε. =1 Ω Since µ a is bounded below with a positive constant C µ, we have µ a δµ s C µ in D with δ = 0, while µ a δµ s C in 3D for soe positive constant C when h is sall enough. Then we apply Holder s inequality to the right hand side of (3.1) such that w Ω D ε D ε. =1 Ω

11 ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION 11 As a result, we have (3.13) ε C D for soe positive constant C. Moreover, fro Lea 1 and, we have D = O(h ). Thus, φ Φ = O(h ). The boundary ter M =1 w Ω ŝ ˆnε becoes ore coplicated for reflection boundary condition (1.3). For siplicity, let us first assue that for each incoing direction ŝ on the angular esh there is a corresponding outgoing direction ŝ on the angular esh with the following reflection condition (3.14) ŝ ˆn = ŝ ˆn, φ = r φ, Φ = r Φ, 0 r 1. Then (3.15) M w =1 Ω ŝ ˆnε = Ω :ŝ ˆn 0 (w w r )ŝ ˆnε, which is analogous to the continuous case. In D, each ter on the righthand side is non-negative since w = w and 1 r 0. In general the above assuption (3.14) is not true. However, if we assue a no perfect reflection condition, i.e., R( x, ŝ, ŝ ) < 1 in (1.3), we can prove the following lea which will be enough for error estiate for general reflection boundary condition in D. Lea 4. If there is no perfect reflection at the boundary, i.e., R( x, ŝ, ŝ ) < 1, r < 1 is true in D. Proof. In D, since the angular esh is unifor, the corresponding reflected angles are also unifor. Moreover, we use linear interpolation between two neighboring angles to interpolate the reflection condition (Figure 3), so at the boundary Ω, we have (3.16) r = α 1 R(ŝ 1, ŝ 1 ) + α R(ŝ, ŝ ), α 1 + α = 1, where ŝ is an outgoing direction that contributes to both incoing directions ŝ 1 and ŝ in the linear interpolation and α 1, α are the linear weights. Fro non perfect reflection condition, we have r < σ < 1. Next we prove the siilar L error estiate for general reflection boundary condition in D. Theore 3 (Reflection Boundary Condition). In D, assue the solution Φ( x, ŝ) to the RTE is C (S 1 ). With no perfect reflection at the boundary, i.e., R( x, ŝ, ŝ ) < σ < 1 in (1.3), and if the angular esh is fine enough, (3.17) φ Φ = O(h ). Proof. The key is to control the boundary ter M =1 w ŝ ˆnε dγ in (3.9). For the general reflection boundary condition (1.3) approxiated by linear interpolation (.8), we have ε ( x) = r ( x)ε ( x) + D ( x), ( x, ŝ ) Γ, ( x, ŝ ) Γ +,

12 1 HAO GAO AND HONGKAI ZHAO where D ( x) is the truncation error due to linear interpolation of the real reflection condition for at the boundary and is of order O(h ). M =1 w Ω ŝ ˆnε = ŝ w ˆn 0 Ω ŝ ˆn ε ŝ w ˆn 0 Ω ŝ ˆn ε = ŝ w ˆn 0 Ω ŝ ˆn ε ŝ w ˆn 0 Ω ŝ ˆn ( ŝ ˆn 0 r ε + D ) Now let us estiate the second ter in the above expression ŝ w ˆn 0 Ω ŝ ˆn ( ŝ ˆn 0 r ε + D ) ŝ w [ ˆn 0 Ω ŝ 1 ˆn σ ( ŝ ˆn 0 r ε ) + 1 D ] 1σ ŝ w [ ˆn 0 Ω ŝ ˆn ŝ ˆn 0 r ε + 1 D ] 1σ ŝ ˆn 0 w Ω ε ( ŝ r ˆn 0 ŝ ˆn ) + 1 1σ ŝ w ˆn 0 = ŝ ˆn 0 w Ω ε (α R 1 1 ŝ 1 ˆn + α 1 R ŝ ˆn ) ŝ w ˆn 0 Ω ŝ ˆn D = ŝ ˆn 0 w Ω ε (O(h ) + R ŝ ˆn ) σ ŝ ˆn 0 w Ω (1 + O(h))σ ŝ ˆn ε + 1 1σ 1σ ŝ ˆn 0 w ŝ ˆn 0 w Ω ŝ ˆn D Ω ŝ ˆn D Ω ŝ ˆn D, where R := R( x, ŝ, ŝ ), the first and second inequalities are fro Jensen s inequality due to the convexity of the function f(x) = x, the second equality is due to linear interpolation of the sooth function, i.e., α 1 + α = 1 as shown in Figure 3, w s are all equal, and r < σ fro Lea 4. When h is sall enough we can find σ < σ < 1 such that M w ŝ ˆnε (1σ ) w ŝ ˆn ε 1 w ŝ ˆn 1 σ D =1 Ω ŝ ˆn 0 Fro (3.1) we have C ε 1 C D Cobining the above results we get (1σ ) w ŝ ˆn ε +C ε 1 1 σ ŝ ˆn 0 Ω Ω ŝ ˆn 0 w Ω ŝ ˆn 0 Ω ŝ ˆn D + 1 C D Since both truncation errors in angular discretization for boundary condition and scattering operator are of order O(h ), we have proved the result. The coplication for reflection boundary condition in 3D ainly coes fro (1) the difficulty for obtaining the siilar inequality as (.9) in soe pathological case when ŝ is alost tangential to the boundary at x, and () the non-unifority of the angular esh, i.e., the failure for establishing the siilar clai in Lea Spatial discretization by DG. In this section, we give an error estiate for the spatial DG discretization of coupled transport equations (.3). First we give a brief suary on DG ethod and its analysis on the following single transport equation on a bounded doain Ω with boundary condition on inflow part of the boundary Ω = { x Ω : ˆn( x) ŝ < 0} { ŝ u( x) + c( x)u( x) = f( x) in Ω (3.18), u = g on Ω where ˆn( x) is the outward unit noral to Ω at x.

13 ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION 13 People have been studying the DG ethod for the convection equation (3.18) since 1974 with the first error estiate on general eshes [7], i.e., (3.19) u u h Ch k u H k+1, followed by the iproved convergence result [] in 1986 with (3.0) u u h Ch k+1/ u H k+1, where k is the degree of polynoial approxiation in each eleent. Actually the result was proved with a stronger nor than L nor. Besides, L p stability and error estiates for p were analyzed for piecewise linear DG on unifor or piecewise unifor triangulation []. However, since the sign of c( x) is undeterined, a special auxiliary function has to be introduced to control the possible exponential growth and get the error estiate. The construction of such an auxiliary function requires convexity of the doain Ω. Two years later, assuing the conforing triangulations that are obtained with slabs of parallelogras divided into two triangles, [3] showed an optial convergence estiate (3.1) u u h Ch k+1 u H k+. Recently, [1] proved a siilar result (3.) u u h Ch k+1 u H k+1. for eshes that are not necessarily conforing or subject to any unifority condition, however ade of siplexes of a unique outflow face. Although we need to deal with a syste of coupled transport equations (.3), the coupling is not very strong. In particular due to the absorption, µ t µ s = µ a > 0, which is also a crucial condition for the stability of the original RTE (1.1), our proof is relatively siple. For the notation convenience, we use h(h a ) to represent the spatial (angular) esh size hereafter. Let us start with the DG forulation of RTE (.3) with vacuu boundary condition. Given a triangulation T h with h = sup K Th dia(k) and the finite-diensional space Vh k which is coposed of functions that are polynoials of at ost k degrees on each K T h, let (u, v) K = K uv and < u, v > B= uv ŝ ˆn B with u K = (u, u) K and << u >> B =< u, u > B, where B Γ h Ω with Γ h = ( k T h K)\ Ω. For each angular direction, let Ω (+) = { x Ω : ˆn( x) ŝ (>)0}. Then DG approxiation φ h (Vh k)m of the solution of (.3) satisfies (3.3) A (φ h, v) = (q, v)+ < q b, v > Ω, for all v V k h, for each, where (3.4) A (φ h, v) = K (ŝ φ h +µ t φ h µ s w φ h, v) K+ < φ h + φ h, v + > Γ + < φh h, v > Ω with φ h ± = liɛ 0 ±φ h ( x ± ɛŝ ) and < u, v > Γ h = Γ h uv ŝ ˆn. Here q and q b represents the internal and boundary source respectively. Note that we can replace φ h by the exact solution φ in (3.3). That is we have the following consistency relation (3.5) A (φ φ h, v) = 0, for all v V k h, for each,

14 14 HAO GAO AND HONGKAI ZHAO Notice that (3.3) can be regarded as a syste of (3.18) coupled by the scattering ter. Therefore, (3.3) can be analyzed siilarly as the vector for of (3.18) with non-negative c, once we can control the scattering ter. This is exactly how it will be carried out in the proof for Theore 4. Here we assue the shape-regularity condition on the esh, i.e., there is a constant σ > 0 such that h K /ρ K σ for each siplex K T h, where h K is the diaeter of the siplex K and ρ k is the diaeter of the largest ball included in K. Theore 4 (Vacuu Boundary Condition). With vacuu boundary condition, when the triangulation T h satisfies the shape-regularity condition and h, h a is sall enough, the error between DG solution φ h via (3.3) and the exact solution φ of (.3) with inflow boundary condition satisfies (3.6) φ φ h Ch k+1/ φ k+1, where the nor is defined by (3.) and φ k+1 := M =1 w φ H k+1. Proof. Let φ be the L projection of φ into V h, i.e., (3.7) (φ φ, v) K = 0, v V k h. Then let s consider (3.3) for e = φ φ h. Using the consistency relation (3.5), we have the following with ẽ = φ φ, (3.8) A (e, e ) = A (e, ẽ ). We first estiate A (e, e ). Since (ŝ e, e ) K = (e, ŝ e ) K + K ŝ ˆne, we have (ŝ e, e ) K = 1 < e+ +e, e + e > Γ h 1 << e >> Ω +1 << e >> Ω. + K As a result, (3.9) (ŝ e, e ) K + < e + e, e + > Γ h + << e >> Ω = 1 << e+ e >> Γ +1 h << e >> Ω, K where < u, v > Ω = Ω uv ŝ ˆn. The scattering ter can be estiated as follow. J := (3.30) w K (µ te µ s w e, e ) K w [ Ω µ te 1 w Ω µ s(e + e )] On one hand, (3.31) w w Ω µ s e = w Ω µ s e. On the other hand, using Lea 3, T := w w Ω µ se (3.3) = (w w + w w O(h a )) Ω µ se w Ω µ se (1 + O(h a))

15 ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION 15 For any h a, there exists a constant δ = O(h a ) such that (3.33) T (1 + δ) w µ s e Ω Cobining (3.30) to (3.33), we have (3.34) J C e where C = C µ δ µ s with lower bound C µ of µ a, and is nonnegative when h a is sall enough in 3D. Notice that C = C µ in D due to rotation invariance of angular esh. Therefore, (3.35) [ w A (e, e ) D e + ] w (<< e + e >> Γ + << e >> h Ω ), where D = in(c, 1 ). Next, let s estiate A (e, ẽ ) and it goes as follow. For the boundary ter, we have (3.36) < e + e, ẽ + > Γ h + < e, ẽ > Ω D 4 (<< e+ e >> Γ + << e >> h Ω ) + 1 D (<< ẽ+ >> Γ + << ẽ >> h Ω ). For the convection ter, using the property (3.7) for L projection, (3.37) (ŝ e, ẽ ) K = (ŝ ẽ, ẽ ) K = 1 ŝ ˆnẽ Siilarly as before, the scattering ter can be estiated as follow, (3.38) w K (µ te µ s w e, ẽ ) K w [ K ( D 8 e K + µt D ẽ K )] + w [ K ( D 8 e K + µs D w ẽ K )] D 4 e + C ẽ Last, fro the standard estiates for L projection, we have (3.39) ẽ K Ch k+1 K φ H k+1(k) ẽ K Ch k+1/ K φ H k+1(k) With the shape-regularity assuption, fro (3.36) to (3.39), we have (3.40) [ w A (e, ẽ ) D e + w (<< e + e >> Γ 4 + << e >> h Ω ]+Ch ) k+1 φ k+1. Cobining (3.8), (3.35) and (3.40), we have shown that (3.41) e Ch k+1/ φ k+1. In the proof C always denotes soe constant. Reark 3.. Cobining estiate (3.1) for angular discretization and (3.6) with k = 1 for our piecewise-linear DG spatial discretization, we have the following estiate for our schee (3.4) Φ φ h = O(h a) + O(h 3/ ). K

16 16 HAO GAO AND HONGKAI ZHAO However second order accuracy is observed in both angle and space in the nuerical tests in [4]. Reark 3.3. The proofs with O(h k+1 ) convergence in [1, 3] do not apply here since the required property on the spatial esh in general is not satisfied for RTE with all possible angular directions. For general reflection boundary condition (.8), the DG forulation (3.4) becoes (3.43) A (φ h, v) = K (ŝ φ h + µ t φ h µ s w φh, v) K + < φ h + φ h, v + > Γ h + < φ h r φ, v > Ω, where r = R(ŝ, ŝ ) 1 and ŝ s are the neighboring directions used in the linear approxiation to approxiate the outgoing direction ŝ that is reflected into the incoing direction ŝ. Now let s prove the sae result as in Theore 4 for general reflection boundary condition in D. Theore 5 (Reflection Boundary Condition). In D, assue that the triangulation T h satisfies the shape-regularity condition and h, h a is sall enough, the error between DG solution φ h via (3.3) and the exact solution φ of (.3) with reflection boundary condition (.8) satisfies (3.44) φ φ h Ch k+1/ φ k+1, if R( x, ŝ, ŝ ) < σ < 1 (no perfect reflection), where the nor is defined by (3.) and φ k+1 := M =1 w φ H k+1. Proof. Following the sae coputation, (3.9) becoes (3.45) K (ŝ e, e ) K + < e + e, e + > Γ h + < e r e, e > Ω = 1 << e+ e >> Γ h + 1 << e >> Ω < r e, e > Ω Now we estiate the last ter in the above expression. < r e, e > Ω 1 ( ) r e + r e ŝ ˆn Ω Following the sae arguent in the proof of Theore 3, when h a is sall enough, there exists a constant σ < σ < 1 such that r ŝ ˆn σ ŝ ˆn. At a point of the boundary where ˆn is well defined, if we su up all angles that satisfy ŝ ˆn 0 in the above expression, ŝ will go over all angles ŝ ˆn 0. We have :ŝ ˆn 0 w ( r e + r ) e ŝ ˆn σ :ŝ ˆn 0 w e ŝ ˆn + σ :ŝ w ˆn 0 e ŝ ˆn σ w e ŝ ˆn Therefore, w < r e, e > Ω 1 σ w << e >> Ω.

17 ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION 17 Hence we get an estiate siilar to (3.35). (3.46) [ w A (e, e ) D e + w (<< e + e >> Γ h +1 σ << e >> Ω ) ] Now we estiate for A (e, ẽ ). Again the only coplication coes fro the boundary ter. We have < e r e, ẽ > Ω =< e, ẽ > Ω < r e, ẽ > Ω For the first ter, we have < e, ẽ > Ω D(1 σ ) 16 << e >> Ω + 4 D(1 σ ) << ẽ >> Ω. To estiate the second ter, we first estiate point-wisely with respect to the angular variable ( ) :ŝ w ˆn 0 :ŝ ˆn 0 r e ẽ ŝ ˆn w ŝ ˆn ( ) r D(1σ ) 3 e + 8 D(1σ )ẽ D(1σ ) 16 :ŝ ˆn 0 w ŝ ˆn e + 8 D(1σ ) :ŝ w ˆn 0 ŝ ˆn ẽ, when the angular esh is fine enough. Therefore, w < r e, ẽ > Ω D(1σ ) 16 w << e >> Ω + 8 D(1σ ) w << ẽ >> Ω Use the sae estiate as in Theore 4 for all other ters, we get (3.47) [ w A (e, ẽ ) D e + w (<< e + e >> σ Γ +1 << e 4 h >> Ω ]+Ch ) k+1 φ k+1. Cobine (3.46) and (3.47) we have (3.48) e Ch k+1/ φ k+1. Again C always denote soe generic constant in the proof. Siilar as before, the ain difficulty to extend the above proof for the reflection boundary condition in 3D is due to (1) the difficulty for obtaining the siilar inequality as (.9) in soe pathological case when ŝ is alost tangential to the boundary at x, and () the non-unifority of the angular esh on the sphere, i.e., the failure for establishing the siilar clai in Lea Spatial discretization by FDM. Here we only consider D FDM (.4) with θ [0, π ), which can be easily extended to other D cases or 3D cases. Here, let f := ax i,j, f i,j,, the ax nor with respect to both angular and spatial variables. Without loss of generality, assue x = y = h. Then we have the following result on the error estiate of FDM (.4) for (1.1). Theore 6. Assue the solution to (1.1) Φ( x, ŝ) is C (Ω S) and denote e i,j, := φ h i,j, Φ(x i, y j, ŝ ), where φ h i,j, is the nuerical solution fro (.4), then (3.49) e = O(h) + O(h a).,

18 18 HAO GAO AND HONGKAI ZHAO Proof. For non-boundary nodes, e i,j, satisfies (3.50) (a + b + µ t )e i,j, (ae i1,j, + be i,j1, ) µ s w e i,j, = T i,j, where the local truncation error T i,j, = O(h) + O(h a) due to first order upwind difference in space and piecewise linear approxiation in angle (Theore 1). Let e i,j, = e. Since a + b + µ t, a, b and µ s w are all non-negative, and (a + b + µ t ) (a + b + µ s w ) = µ a > 0, at (i, j, ), the linear syste has an M-atrix and we have (3.51) (a + b + µ t ) e a e + b e + µ s w e + T i,j,. Since w = 1, we have (3.5) µ a e T. Let C be the positive lower bound of µ a, thus we have proved the stateent that (3.53) e 1 C T = O(h) + O(h a). For boundary nodes with general discretized reflection boundary condition (.8), if ŝ ˆn > 0, i.e., (x i, y j, ŝ ) Γ +, after we use relation (.8) to substitute all e i,j, with ŝ ˆn 0, we have (a + b + µ t )e i,j, (ae i1,j, + be i,j1, ) µ s ( :ŝ ˆn>0 w e i,j, + :ŝ ˆn 0 w :ŝ ˆn>0 r e i,j, ) = T i,j, Here T i,j, ay include the error due to linear interpolation of the exact reflection boundary condition, which is of order O(h a). Since r 0 and r 1, the M-atrix syste is preserved by our angular discretization at the boundary. The sae arguent used above shows e = O(h) + O(h a). if ŝ ˆn 0, fro boundary condition (.8) we have e i,j, = r e i,j, + T i,j, :ŝ ˆn>0 Again T i,j, ay include the error due to linear interpolation of the exact reflection boundary condition. Cobining the above cases, we see that e = O(h) + O(h a) is also true for boundary nodes. Since the fully discretized linear syste has a M-atrix structure, the ISI (.6), which is a Gauss-Seidel iteration, converges and converges faster than the standard SI (.7). Here is a siple arguent. Let E n := φ n i,j, φh i,j,, by the siilar arguent in Theore 3, for ISI (.6), we have (3.54) E n r M =+1 w 1 + r M =+1 w E n1 = (1 ρ )E n1,

19 ANALYSIS OF A NUMERICAL SOLVER FOR RADIATIVE TRANSPORT EQUATION 19 where r = µ s /µ a and ρ = 1 1+r M =+1 w. Siilarly, for SI (.7), we have (3.55) E n r M =1 w 1 + r M =1 w E n1 = (1 ρ J )E n1. with ρ J = 1 1+r. Obviously, ISI converges faster than SI since ρ > ρ J. When scattering is weak, e.g., r = µ s /µ a 1, or in the case of forwardpeaking regie, which is usually the case in optical iaging, e.g., g 1 in H-G, 1 w 1, it can be seen easily fro the above arguent that the iproveent of ISI over SI is significant. In the extree case with g = 1, w = 0 for, ISI takes only one iteration when the spatial ordering is along the characteristic for each transport direction. When scattering is doinant (r >> 1) and f is isotropic, or the doain is optical thick, even ISI ay converge quite slow. Since ISI captures transport effectively and is a soothing operation, it is used as a natural relaxation schee for a ultigrid algorith in both space and angle [4], which can deal with RTE in different regies effectively. References 1. Cockburn B, Dong B, and Guzán J, Optial convergence of the original discontinuous galerkin ethod for the transport-reaction equation on special eshes, SIAM J. Nuer. Anal. 3 (008), Johnson C and Pitkäranta J, An analysis of the discontinuous galerkin ethod for a scalar hyperbolic equation, Math. Cop. 46 (1986), Richter GR, An optial-order error estiate for the discontinuous galerkin ethod, Math. Cop. 50 (1988), Gao H and Zhao HK, A fast forward solver of radiative transfer equation, Transport Theory and Statistical Physics 38 (009), Jackson JD, Classical electrodynaics, 3rd edition, Wiley, New York (1999). 6. Case KM and Zweifel PF, Linear transport theory, Addison-Wesley, Massachusetts (1967). 7. Lesaint P and Raviart PA, On a finite eleent ethod for solving the neutron transport equation, Matheatical aspects for finite eleents in partial differential equations, Acadeic Press, New York (1974). 8. Reed WH and Hill TR, Triangular esh ethods for the neutron transport equation, Los Alaos Scientific Laboratory Report LA-UR (1973). Departent of Matheatics, University of California, Los Angeles, CA E-ail address: haog@ath.ucla.edu Departent of Matheatics, University of California, Irvine, CA E-ail address: zhao@ath.uci.edu