Comparison of Three Approximations to the Linear-Linear Nodal Transport Method in Weighted Diamond-Difference Form

Transcription

1 NUCLEAR SCIENCE AND ENGINEERING: 1, 19-2 (1988) Comparison of Three Approximations to the Linear-Linear Nodal Transport Method in Weighted Diamond-Difference Form Y. Y. Azmy Oak Ridge National Laboratory, Engineering Physics Mathematics Division P.O. Box 28, Oak Ridge, Tennessee Received October 12, 1987 Accepted June 1, 1988 Abstract Two previously derived approximations to the linear-linear nodal transport method, the linear-nodal (LN) the linear-linear (LL) methods, are reexamined, together with a new approximation, the bilinear (BL) method, that takes into account the bilinear nodal flux moment. The three methods differ in the degree of analyticity retained in the final discrete variable equations; however, they all possess the very high accuracy characteristic of nodal methods. Unlike previous work, the final equations are manipulated cast in the form of the classical weighted diamond-difference (WDDj equations (not just a WDD algorithm). This makes them simple to implement in a computer code, especially for those users who have experience with WDD algorithms. Other algorithms, such as the nodal algorithm, also can be used to solve the WDD-form equations. A computer program that solves two-dimensional transport problems using the LN, LL, or the BL method is used to solve three test problems. The results are used to confirm our algebraic manipulations of the nodal equations also to compare the performance of the three methods from the computational, as well as the theoretical, point of view. The three methods are found to have comparable accuracies for the problems studied, especially on meshes that are sufficiently fine. I. INTRODUCTION Nodal methods have been developed implemented for the numerical solution of the discrete ordinates neutron transport equation. 1-8 Numerical testing of these methods comparison of their results to those obtained by conventional methods have established the high accuracy of nodal methods. 1 ' 2 Furthermore, it has been suggested that the linear-linear approximation is the most computationally efficient, practical nodal approximation. 2 Indeed, this claim has been substantiated by comparing the accuracy of the solution the CPU time required to achieve convergence to that solution by several nodal approximations, as well as the diamond-difference scheme. 2 ' 3 Two types of linear-linear nodal methods have been developed in the literature: analytic linear-linear (NLL) methods in which the "transverse leakage" terms are derived analytically, 1 ' 2 approximate linear-linear (PLL) methods in which these terms are approximated. 2 " 6 In spite of their high accuracy, NLL methods result in very complicated discrete variable equations that exhibit a high degree of coupling, thus requiring special solution algorithms. On the other h, the sacrificed analyticity in PLL methods is compensated for by the simpler form of the resulting discrete variable equations that make possible the use of diamond-differencelike solution algorithms. 3 " 6 One PLL scheme that was previously developed implemented included a bilinear component, i.e., the Pi(Af) - P\{y) Legendre moment, of the node interior expansion of the flux. 2 For the two test problems considered in Ref. 2, this scheme was found to be very similar to an NLL scheme (also presented in Ref. 2). That is, inclusion of the bilinear moment resulted in a high accuracy of the solution, but it resulted in the coupling of the discrete variable equations. 2 Hence, no attempt has been made so far to derive an NLL method that includes the bilinear moment of the flux, as this would have resulted in a

2 very cumbersome set of equations. In addition, it is not clear yet how much can be gained, in terms of better accuracy, at the cost of the anticipated complication of such a method. The purpose of this work is to derive NLL PLL methods, both of which have been derived before, to cast them in a simple weighted diamonddifference (WDD) form. Also, we derive a new NLL method that includes the bilinear (BL) moment of the flux, the BL method, cast in a simple WDD form. This BL method is a special case, i.e., order one of a general order nodal method cast in WDD form that has been derived 9 for transport problems of general dimensionality in Cartesian geometry. This general order method has also been implemented in a computer code 9 ' 1 used to solve a simple test problem with approximation orders up to order nine. It must be emphasized that the final equations for the three nodal approximations presented here are explicit WDD relations, with analytically derived spatial weights, which may be solved via a WDD algorithm. This is in contrast to the nodal equivalent finite difference (NEFD) method 3 in which only the solution algorithm, but not the form of the final equations, resembles WDD. This is an important difference, especially since one of the main criticisms directed toward nodal methods in general is the difficulty of their final forms the complicated algorithms necessary to solve them. Our ability to write the nodal equations in a simple WDD form makes the high accuracy of such methods accessible to the many users familiar with conventional WDD methods, with minor modifications. In Sec. II of this paper, we simultaneously derive the discrete variable equations for the three nodal methods presented here. These equations are shown to form a closed set of algebraic equations (i.e., as many equations as unknowns). In Sec. Ill, we manipulate the discrete variable equations to obtain a simple WDD form of the equations, with spatial weights that are analytically derived. The three linear-linear nodal methods discussed in this paper are compared from a theoretical point of view in Sec. IV from a computational point of view in Sec. V. Also, in Sec. IV, we describe the nodal algorithm the WDD algorithm for each method, as well as the algorithm that we actually used in our computer code, which is a hybrid of the nodal WDD algorithms. A brief summary of the work the main conclusions are presented in Sec. VI. In this section, we present the nodal formalism through which the discrete variable equations are derived from the continuum neutron transport equation. In this development we assume steady state, we consider the case of a monoenergetic, external source problem in two-dimensional Cartesian geometry with isotropic scattering. The resulting methods can be extended easily to many energy groups eigenvalue problems by employing an inner-outer iteration procedure. Addition of a third dimension is straightforward but lengthy. We derive simultaneously the three approximations of the nodal method considered here: the linearnodal (LN) method 2 ' 4 (which is a PLL method), the linear-linear (LL) method, 1 ' 2 the BL method (both NLL methods). The discrete ordinates approximation of the transport equation in two-dimensional Cartesian geometry is given by where Hk- W +1 dx tlc +l (x,y) m i+i + V k dy + (7^+1 = 5' (1) n k, 7) k = x y components of the k'th discrete ordinate (l k, k = I,...,N = (+ 1) inner iterate of the Ar'th angular flux a = total cross section. The 'th inner iterate of the source S 1 normally contains the given external source as well as the in-scattering source calculated from the 'th inner iteration (= o s 4>'\ o s is the scattering cross section <j> 1 is the 'th inner iterate of the scalar neutron flux given by <f> 1 = w k ip l k) updated after every inner iteration. The inner iteration procedure is irrelevant to the remainder of the derivation; hence, the superscript indicating the iteration index is suppressed. First, we divide the domain of the problem into M rectangular computational elements, i.e., nodes, of the form x [~b u +b x ], 1 = 1,...,M. We define Legendre polynomials on the interval [ c,+c] by Pn(z) = P (zc), n>, (2) where P n is the regular Legendre polynomial of order n, defined on the interval [-1,+1]. The normalization relation for p n then becomes 1 r +c, 2 -J dzp m (z)p n {z) = <5 (2m + 1) Then we define the transverse moments of the angular flux by (3) II. THE NODAL FORMALISM FOR THE NEUTRON TRANSPORT EQUATION tk.xjm = th:,iy(y) = 2y + i \ C +b, J dyf*(x,y)pj(y) 2b (4a) 2 i a -a dx^k(x,y)p i (x), (4b)

3 where we have suppressed the node index on the node dimensions a b. The transverse moments of the source S xj (x) S iy (y) are defined analogously. Also, the nodal moments of the angular flux are defined by tk,ij = 2 i+ l\(2j i±i\ r * 2 a 2b IJ +b +t> X (5) -b with analogous expressions for the nodal moments of the source. Now we operate on the transport equation, Eq. (1), by the transverse moments operator; namely, we multiply Eq. (1) by (2i + \2a)pj(x) integrate over the x dimension of the node to obtain the x moments equations Vk-r^kjy + al Pk,iy ~ S iy + Xk,iy» dy where the x leakage moments are given by =,1, (6) Xk.oy(y) s lm+a,y) - M-a,y)] (7a) 2 a Xk,iy(y) = -4 im+a,y) - 2t k, y (y) 2 a + M-a,y)] ( 7b ) The same procedure is repeated in the y direction to obtain analogous y moments equations expressions for the y leakage moments. This set of equations is exact, albeit not closed. Closing the system requires exping the surface fluxes on the right sides of Eqs. (7) its y analogue in a truncated series. Since the order of this expansion is different for each of the three methods considered here, we introduce the coefficients Vj =, if i = 1, LN or LL method = 1, otherwise, (8a) X, =, if = 1, LN method = 1, otherwise. (8b) Both the zeroth first-order moments of the leakage terms are retained in the LL BL methods, while in the LN method, the bilinear component of the leakage terms is neglected. That is, for the three methods Eq. (7) are approximated by Xk,Oy(y) = ^ Wk.xo(+fl) - tk.xo(-a)] 2 a - [ t f - * k, x i ( - a ) ] i (9a) 2 a b -3Xt Xk \yiy) = -Z [ik,xo(+a) - Wk, + tk,xo(-fl)] 2a Xi^ [tk,xi (+«) + tk,xa~a)] 2 a (9b) The next step is to substitute Eqs. (9) into the right sides of Eqs. (6), then solve the latter exactly by using an integrating factor. Evaluating the solution at the node surface, we obtain exp{e k )\p k, y (+b) = (2 sinhe^) - exp(-e k )\l kyqy (-b) 3 oo r r Wk,xo(+o) 2 6 k - tk,xo(~a)] + 2 coshe^ - smiley Ck J_ x - - -^-Wk.xd+a) - tk,x l(-a)] 25k exp(e k )\p kay (+b) - exp(-e k )\p kay where (-b) = (2sinhe*) j - -^r W k, x o(+ a ) ~ 2 tk,oo 26k + 2 COShe* sinhe^- k + fk,xo(~a)] S 3 A X 1 fi - T7 1 Wk,xi(+a) - 2^,1 a 2 dk 8 k = aa^k e k = abr} k. (1a) (1b) Two analogous equations can be derived from the y moments equations. Note that Eq. (1b) contains the second difference between the three methods considered in this paper. Because neither the LN or the LL method takes into account the bilinear moment of the flux, the bilinear component of the source S n, arising from in-scattering is neglected in these two methods. In contrast, in the BL method (v t = 1) the bilinear component of the source is calculated every iteration. Equations (1) their y moments analogue represent a total of four equations. In addition, there are

4 four incoming flux moments boundary conditions relating the surface flux moments in a given node to those in an adjacent node or to the global boundary conditions. On the other h, the list of discrete variable unknowns includes eight surface flux moments three (four) nodal flux moments for the LN LL (BL) methods. The source moments have not been counted as unknowns because in every iteration they are explicitly written in terms of the known nodal flux moments calculated in the previous iteration. Hence, it is clear that additional relations between the discrete variable unknowns are necessary in order to solve the set of equations uniquely. These relations are obtained by requiring the nodal balance of all retained flux moments in each method. That is, we multiply Eq. (1) by 2i+ 1\ (2j+ 1 2 a 2b Pi(x)pj(y), 'J =,1, integrate over the node to obtain 28, ltk,xo(+a) ik,x o(-a)] e k X [fk,y( + b ) ~ tk,y(~b)] + ^ = -> etc. (11) The equation resulting from the case i = j = 1 is valid only for the BL method where the bilinear moments are calculated conserved; thus, the four balance equations close the system of algebraic equations for the BL method. In the LN LL methods, the set of algebraic equations is closed without the i = j = 1 equation, which is not valid anyway since the bilinear moments are not calculated in these methods. The set of algebraic equations represented by Eqs. (1), their y moments analogues, Eqs. (11) can be solved in their present form using the traditional process of sweeping the mesh. However, this results in complicated expressions requires the storage or recalculation of many coefficients for each node type each distinct angular direction. Instead, in Sec. Ill we manipulate these equations in order to cast them in a simple WDD form that requires the storage of only one (two) spatial weight for each node type per distinct angular direction for the BL (LN LL) method. III. THE WDD FORM based on simultaneously solving the equations for the outgoing flux moments [e.g., for n k 5, \j kixj (±a), etc.] first, then using these to calculate the nodal flux moments, which contribute to the new iterate of the scalar flux. In the WDD algorithm 3,4 the nodal flux moments are evaluated first from uncoupled expressions, then used to calculate the outgoing flux moments. The advantage of the WDD algorithm is that it does not require the simultaneous solution of algebraic equations in each node. This is even more important in three-dimensional problems where the number of coupled equations to be solved simultaneously in the nodal algorithm is larger. However, the WDD algorithm results in very complicated expressions 3 has been limited so far to the LN method. 3 ' 4 Our objective in this section is to rewrite the discrete variable equations in the form of WDD relations to which either the nodal or WDD algorithm may be applied. By a WDD relation we mean an expression for the nodal flux moments in terms of the surface-evaluated transverse flux moments, that is, fully specified by a spatial weight. The spatial weights in the present work are not preassigned, but rather derived analytically from the equations obtained via the nodal formalism. Also, the form of a WDD relation is modified here to accommodate the presence of the first-order flux moments that are not commonly incorporated in conventional WDD methods. To obtain the WDD equations, we eliminate the source moments from Eqs. (1) using the full set of Eqs. (11). This yields four expressions for each method, relating the nodal flux moments to the transverse flux moments. The coefficients appearing in these equations can be written in terms of the spatial weights defined by = Oik,i to become cothsi (cothsj. - h \ cothet Vil. cothe^ - e* V h)\ (12a) =,1 (12b) Two algorithms have been used previously to solve the discrete variable equations, Eqs. (1) (11), for the LN LL methods. The nodal algorithm 1 is + fik,lpk,l (13a)

5 I +Pk.i tk,ly(+b) + ^-^ykayi-b) TABLE I Summary of Comparison Between the Three Linear-Linear Methods where = tk, 1 + VlPk,ltk, 11 3X 2b k X = Xi(l - i'i) = 1, if LL method (13b) =, otherwise. (14) Two equations analogous to Eqs. (13), with the x transverse flux moments appearing on the left side, can also be derived. This set of four equations now replaces Eqs. (1) their x moment analogues are solved simultaneously with Eqs. (11), using the nodal or the WDD algorithm. In Sec. IV, we compare the equations for the three methods from the theoretical point of view; then in Sec. V we present numerical results for test problems solved by the three methods to compare their performance. It is worthwhile noting that while the LN LL method formalisms are not new, have been derived implemented before, 1 " 6 writing their final equations in the simple WDD form of Eqs. (13) is new. IV. COMPARISON OF THE THREE LINEAR-LINEAR NODAL METHODS In the absence of a complete rigorous error analysis of nodal transport methods in general, the three methods considered here in particular, choosing a certain method to solve a given problem is often based on intuition experience. From the comparisons presented in this the following section, this author is convinced that none of the methods dominates the other two. That is, for each problem to be solved a compromise has to be reached between the available resources (such as the memory size, computer speed, etc.) the achieved results (such as accuracy, simplicity of method, solution algorithm, etc.). The main differences between the three methods are summarized in Table I are discussed below in more detail. IV.A. The LN Method For the LN method there are two distinct spatial weights per dimension per distinct discrete ordinate for each node type. This is the simplest of the three methods because it has the least coupling between the LN LL BL Number of weightsordinate dimensionnode Number of discrete variable unknownsordinatenode Include bilinear flux moment No No Yes Include leakage bilinear component No Yes Yes Coupling of WDD equations in different dimensions No Yes No WDD algorithm possible Yes Yes Yes Number of equations solved simultaneously in WDD algorithm Maximum coupling 3 in WDD equations Minimum coupling 3 in WDD equations Maximum coupling 3 in balance equations Minimum coupling 3 in balance equations a The maximum (minimum) coupling in a set of equations is the maximum (minimum) number of unknown variables appearing in any equation in the set. algebraic equations; however, it contains the least analytical information. In fact, as in the NEFD algorithm, 3 the LN method equations can be manipulated to obtain uncoupled expressions for the nodal flux moments in terms of the incoming transverse flux moments. An interesting fact about the expressions for the spatial weights, Eqs. (12) is that they are odd functions of the discrete ordinate. For example, = a ki (ii k ). This significantly reduces the storage requirement simplifies the solution procedure by allowing us to write the final set of equations in terms of incoming outgoing transverse flux moments, which is required by the sweep methodology. For example, for \j, k 5, the outgoing transverse flux moment \l k \ x j = \j ktx j(±a), the incoming transverse flux moment \l k, x j =,*,,(+#) The balance equations then become [$k,x - Vk.xo] + r [t?k,y - ^k,y] + tk.oo 2b k le k SQOO, (15a) 3 s x 1 7TT Wk.xO - 2^, + i'k,xo\ + -Z Wk,ly ~ y\ lb k le k + tk, 1 = Sioff, (15b)

6 lb k [Kxl - + 7T Wk,y - 2^, + t'k,y] le k + tk,o i=s 1 a. (15c) The WDD relations become = ''Ml le k s y X Wk,ly ~ 2^,1 + Vk,\y\. (17b) = tk, + Sx<Xk,otk,io (16a) + =, d6b) two analogous equations for the y moments, where = sign s y = sign(r? A: ), h k, e k, a kj, P kii represent the absolute values of the corresponding quantities. Equations (16a) (16b) are equivalent to Eqs. (2) (4b) of Ref. 4, respectively. The nodal algorithm is obtained by using Eqs. (16) their analogues to replace the nodal flux moments in Eqs. (15). In addition, an equation expressing the uniqueness of it:,oo> tk,oi> an d io is obtained by eliminating these quantities from Eqs. (16) their analogues, yielding a constraint equation involving only the outgoing flux moments. This results in a set of four coupled equations that must be solved simultaneously for the outgoing flux moments in terms of the incoming flux moments. Alternatively, a WDD algorithm [equivalent to the NEFD (Ref. 3)] is obtained by solving Eqs. (16) their analogues for the outgoing flux moments, then substituting the result in Eqs. (15). This yields three equations that can be solved for the nodal flux moments before they are coded or can be solved numerically for each node. In the code we developed, for which the results presented in Sec. V were obtained, we chose to implement the second option for its simplicity. IV. B. The LL Method The LL method also has two distinct spatial weights per dimension, per distinct ordinate, per node type. It has a higher degree of coupling than the LN BL methods, which may explain why a WDD algorithm has never been implemented for it. The LL method retains more analytical information in the discrete variable equations than the LN method, manifested in the higher order representation of the leakage terms. The nodal balance equations are the same as Eqs. (15). The WDD equations are two analogous equations in the y moments. Just like the LN method, a nodal algorithm a WDD algorithm are possible here. To obtain the nodal algorithm, Eq. (17b) its analogue are solved for t'ar.io! then i^ qo is obtained from the analogue of Eq. (17a), yielding four coupled equations in the four outgoing flux moments. On the other h, the WDD algorithm is obtained by individually solving Eq. (17a) its analogue for \[^x \l k,oy, then simultaneously solving Eq. (17b) its analogue for \l kx \ \p kt i y. The resulting expressions are substituted in the balance equations, yielding three coupled equations that must be solved simultaneously for the nodal flux moments. In our implementation of this method, we analytically eliminated yp^x yj kfiy using Eq. (17a) its analogue, then numerically solved the remaining five equations in each node. IV. C. The BL Method The BL method is unique (as is the general order nodal method presented in Ref. 9) in that it requires the calculation storage of only one spatial weight per dimension per distinct discrete ordinate for every node type. Of course, it contains the most analytical information in the algebraic equations. It has more coupling than the LN method, but less than the LL method. However, it requires the calculation of an additional angular flux moment the storage of an additional scalar flux moment. In the context of the three test problems of Sec. V, there seems to be very little gain in accuracy due to using the BL method as opposed to the LN or LL method. Problem configurations may occur, however, which lead to bilinear moments of relatively large magnitude that would cause the BL method to be more accurate than the otherwise almost identical LN LL methods. For the BL method, the balance equations, Eqs. (15), must be augmented by a conservation equation for the bilinear nodal flux moment: = ^Ar.oo + s x a k,o^k,io (17a) + = Snff. (18)

7 The WDD relations are given by = ik, + s xa k\p ky 1 (19a) 1 ^ *, <T t = 2 <J T = ' 2 VACUUM V 2 <7 S =.1 a s =.1 s = s = 2 _L = tk,1 + s x a k \p k<n, (19b) with analogous equations for the y moments. By similar arguments as before, one can derive both a nodal a WDD algorithm for the BL method. The algorithm we adopted in our code was derived by solving Eqs. (19) their analogue for the outgoing flux moments, which are then eliminated from the balance equations, Eqs. (15) (18). This yields four coupled equations in four nodal flux moments, which are solved simultaneously in every node. 's y I a T= 1 <x T = 2 o s =.5 o s =.1 s = 1 s = 1 1! 5 1 Fig. 1. Geometry nuclear data for test problem 1. II V. NUMERICAL COMPARISON OF THE THREE LINEAR-LINEAR NODAL METHODS We have implemented the equations for the LN, LL, BL methods derived above in a computer code using the algorithms described in Sec. IV to solve them numerically. These are, in some sense, a combination of the nodal WDD algorithms; the final equations are coupled (like the nodal algorithm) are solved simultaneously for the nodal flux moments (like the WDD algorithm). This was preferred over a pure WDD algorithm because it is much simpler can be extended easily to three-dimensional geometry. To monitor easily the memory size required to solve a problem, we used a container array methodology, in which all arrays have variable lengths are contained in one array of fixed length, the container array A. Three test problems were solved using the three methods in order to compare several of their computational aspects, also to validate our derivation of the WDD form. The first test problem is shown in Fig. 1. This problem was solved on a sequence of meshes using an S 4 EQN-type angular quadrature, a pointwise, relative convergence criterion of 1~ 4 on each one of the calculated nodal flux moments. The converged solution was used to calculate the quadrant-averaged scalar fluxes flux averages over the two regions denoted 1 2, shown in Fig. 1. This was to make possible comparisons of both global quantities as well as relatively local quantities. The results of the calculations are shown in Table II. Also shown is the length of the container array A, the number of iterations CPU time (in seconds) required to achieve convergence, the number of calculated negative fluxes. In general, the LN method takes the shortest CPU time to converge, followed by the BL method, while the LL method takes longest. This is not surprising since in the algorithm we adopted, three, four, five equations, respectively, are solved simultaneously for each direction per node. Of course, the BL method consumes more memory than the other two methods because it requires the storage of two additional arrays containing the old new bilinear moments of the scalar flux. Therefore, in problems with only a few material regions, such as the three cases presented here, the BL method requires more storage; however, for problems with many material compositions, or high-order angular quadrature, hence many distinct spatial weights, the BL method may require a smaller storage area due to the fewer spatial weights necessary. The calculated quantities seem to change only very little with the method, especially at locations where the flux is relatively large. In fact, several quantities calculated by the three methods on the same mesh are identical. Moreover, all three methods yield approximately the same number of negative scalar fluxes. The second test problem is a modification of Khalil's steel water problem 11 with a smaller average scattering ratio to avoid excessively slow convergence since our code currently is not equipped with acceleration methods. The geometric configuration nuclear data are shown in Fig. 2. We used an S 4, EQN-type angular quadrature with a 1~ 4 pointwise relative convergence criterion on all the calculated nodal moments of the scalar flux by each method. The

8 TABLE II Comparison of the Numerical Solutions by the Three Methods for Test Problem 1 (see Fig. 1)* Number of Length Number of CPU Negative Mesh I II IV 1 2 of A Iterations (s) Fluxes LN Method 1 X E l a.1986e E X E 1.199E E x E E E LL Method 1 x E E E x E E E x E E E BL Method 1 x E E E x E E E X E E E ^Calculations were performed on the Vax 86 at Oak Ridge National Laboratory (ORNL). a Read as.417 x 1"'. 1i 5' 4' VACUUM V (7y. 1, a s =.1 s = III! 1 Oj =.3, a s =.1 L s = II Oj.1, <7S =.1 -s = I <r T = 1 a s =.5 s = 1 i : 2» Fig. 2. Geometry nuclear data for test problem 2. results for this problem presented in Table III suggest conclusions identical to those reached from the results of the first test problem. Here, the difference between the solutions obtained by the three methods is more pronounced especially at low flux regions when coarse meshes are used. However, the solutions seem to converge to one another when finer meshes are used. The third test problem is a monoenergetic version of the well-logging problem. 3 ' 12 The geometry nuclear data for this problem are shown in Fig. 3. We solved this problem using an S 6, EQN-type angular quadrature with a 1~ 5 pointwise relative convergence criterion on all the calculated nodal moments of the scalar flux by each method. The converged solution in each case was used to calculate the average flux in regions D\ D 2, which represent the locations of the detectors usually used in this configuration. The results are presented in Table IV, where we also include the results for the same problem using the constant-linear (CL) the constant-quadratic (CQ) methods. 13 The LN LL methods results shown in Table IV were obtained from Ref. 13 were also identically calculated using our computer code. This is evidence of the correctness of the derivation of our WDD form, at least for these two methods. Furthermore, this numerically verifies the mathematical equivalence that we claim between the WDD form previous implementations of the LN LL methods. The two finest meshes' results for each method (except the CQ method) were used to obtain h 2 -extrapolated values for the detector responses; for all methods these were for D x.1247 x 1 _1 for D 2. The percent error in each calculated quantity

9 TABLE III Comparison of the Numerical Solutions by the Three Methods for Test Problem 2 (see Fig. 2)* Length Number of CPU Mesh I II III IV 1 2 of A Iterations (s) LN Method 1 x E-l a.2222e E E X E E E E X E E E E X E E E E LL Method 1 x E E E E x E E E E X E E E E x E E E E BL Method 1 x E E E E X E E E E x E E E E x E E E E Calculations were performed on ORNL's Vax 86. a Read as.1626 X 1"'. o D o < > O h- C3 WATER VACUUM l- 1 LIMESTONE (shown in parentheses in Table IV) is then obtained as the percentage relative difference between that quantity the corresponding extrapolated value. Comparison of the errors of the different methods suggests that for this problem the LN, LL, BL methods are more accurate, on the same mesh, than the CL CQ methods. The puzzling aspect of this comparison lies in the fact that the LN LL methods appear to be more accurate than the BL method, even though the latter has fewer approximations made in its formal development than either the LN or the LL method. It is comforting, however, that all three methods produce the same results, which are also identical to the extrapolated values when the mesh is refined VI. SUMMARY AND CONCLUSIONS 8 - S = 1 ' Fig. 3. Geometry for the well-logging problem. 3 The nuclear data are as follows: c T (a s ) for limestone = ( ); for water = ( ); for steel = (.49446) cm" 1. We have rederived the LN the LL methods for solving the neutron transport equation, which have been shown previously to possess very high accuracies on coarse meshes. Our main objective was to simplify the final equations for these methods whose difficulty limited the widespread practical use of high-order nodal methods. We were able to write the final equations for these two methods, as well as a third new method, the BL method, in a simple WDD form.

10 TABLE IV Comparison of the Numerical Solutions by the Three Methods as Well as the CL CQ Methods 13 for the Well-Logging Problem (see Fig. 3)* Mesh A D 2 (x 1) Length of A CPU (min) Number of Iterations Number of Negative Fluxes CL Method 13 7x8 14 x x X 64 7x8 14 x x (-36.9) (-8.5) 1.68 (-2.2) 1.78 (-.52) (-21.8) 1.63 (-6.6) (-2.).7639 (513).1112 (-1.8).1212 (-2.8).1238 (-.72).9339 (649).1142 (-8.4).1215 (-2.6) CQ Method 13 LN Method 7x8 14 x x X (-4.1) (.87) (.6) (1.2).126 (1.) A LL Method 7x8 14 x x X (-6.3) (-.12) (-.6) (1.36).1242 (-.4).1246 (-.8) BL Method 7x8 14 x x X (-15.5) (-1.9) (-.17) (-18.8).1214 (-2.6).1243 (-.32) "Calculations were performed on ORNL's VAX 86. The LN method is the simplest of the three methods. The bilinear term in the scattering source is neglected, the expansion of the transverse leakage term in the linear moments equation is truncated at the zeroth order. Yet, the LN method has been successfully applied by other authors to obtain accurate solutions to transport problems. In the LL method, the transverse leakage is exped in a linear series in the equations for the constant linear moments. Hence, it contains more analytic information but is far more complex than the LN method. The accuracy of the LL method also has been demonstrated elsewhere. The BL method differs from the LL method only in that it retains the bilinear moments of the scattering source. Of the three methods, it is the most complete analytically, at the expense of calculating storing an additional nodal flux moment, the bilinear moment. On the other h, its WDD form is written in terms of only one spatial weight (versus two for LN LL) per distinct discrete ordinate for each node type. Furthermore, the BL final equations are less coupled than the LL equations, which makes their numerical solution faster, in spite of the additional effort expended in calculating the additional flux moment. We coded the three methods in one computer program used them to solve three test problems in order to verify the derivations to compare the computational performance of the three methods. The numerical results indicate that the solutions to the three methods are very close to one another, especially on fine meshes. The three methods seem to differ most at locations of very low flux levels. It is very difficult

11 to reach rigorous conclusions from the results of a few test problems. Additional work on the error analysis of these methods will be necessary to identify classes of problems to which each method is more suitable to apply. ACKNOWLEDGMENTS Part of this work was done while the author was at the University of Virginia, Charlottesville, Virginia. This work was sponsored by the U.S. Department of Energy under contract DE-AC5-84R214 with Martin Marietta Energy Systems, Inc. REFERENCES 1. R. D. LAWRENCE J. J. DORNING, "A Discrete Nodal Integral Transport Theory Method for Multidimensional Reactor Physics Shielding Calculations," Proc. Conf. 198 Advances in Reactor Physics Shielding, Sun Valley, Idaho, September 14-19, 198, p. 84, American Nuclear Society (198). 2. W. F. WALTERS R. D. O'DELL, "Nodal Methods for Discrete-Ordinates Transport Problems in (x,y) Geometry," Proc. Topi. Mtg. Advances in Mathematical Methods for the Solution of Nuclear Engineering Problems, Munich, FRG, April 27-29, 1981, Vol. 1, p. 115, American Nuclear Society (1981). 3. A. BADRUZZAMAN, Nucl. Sci. Eng., 89, 281 (1985). 4. W. F. WALTERS, Nucl. Sci. Eng., 92, 192 (1986); see also W. F. WALTERS, "Augmented Weighted Diamond Form of the Linear Nodal Scheme for Cartesian Coordinate Systems," Proc. Topi. Mtg. Advances in Nuclear Engineering Computational Methods, Knoxville, Tennessee, April 9-11, 1985, Vol. 2, p. 452, American Nuclear Society (1985). 5. A. BADRUZZAMAN, Z. XIE, J. J. DORNING, J. J. ULLO, "A Discrete Nodal Transport Method for Three-Dimensional Reactor Physics Shielding Calculations," Proc. Topi. Mtg. Reactor Physics Shielding, Chicago, Illinois, September 17-19, 1984, Vol. 1, p. 17, American Nuclear Society (1984). 6. R. L. CHILDS W. A. RHOADES, Trans. Am. Nucl. Soc., 5, 476 (1985). 7. J. DORNING, "Nodal Transport Methods After Five Years," Proc. Int. Topi. Mtg. Advances in Nuclear Engineering Computational Methods, Knoxville, Tennessee, April 9-11, 1985, Vol. 2, p. 412, American Nuclear Society (1985). 8. R. D. LAWRENCE, Prog. Nucl. Energy, 17, 271 (1986). 9. Y. Y. AZMY, Nucl. Sci. Eng., 98, 29 (1988). 1. Y. Y. AZMY, Trans. Am. Nucl. Soc., 55, 35 (1987). 11. H. KHALIL, Nucl. Sci. Eng., 9, 263 (1985). 12. Y. Y. AZMY, Trans. Am. Nucl. Soc., 54, 187 (1987). 13. R. D. LAWRENCE, IBM, Private Communication (Sep. 1986).