A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical and spherical coordinates

Transcription

1 A formally fourth-order accurate compact scheme for 3D Posson equaton n cylndrcal and sphercal coordnates Mng-Chh La Ju-Mng Tseng Department of Appled Mathematcs Natonal Chao Tung Unversty Hsnchu 300, TAIWAN Abstract In ths paper, we extend our prevous work (M.-C. La, A smple Compact Fourth-Order Posson Solver on Polar Geometry, J. Comput. Phys., 182, (2002)) to three-dmensonal cases. In partcular, we present a spectral/fnte dfference scheme for Posson equaton n cylndrcal and sphercal coordnates. The scheme reles on the truncated Fourer seres expanson, where the partal dfferental equatons of Fourer coeffcents are solved by a formally fourth-order accurate compact dfference dscretzaton. Here the formal fourth-order accuracy means that the scheme s exactly fourth-order accurate whle the poles are excluded and s thrd-order accurate otherwse. Despte the degradaton of one order of accuracy due to the presence of poles, the scheme handles the poles naturally; thus, no pole condton s needed. The resultant lnear system s then solved by the B-CGSTAB method wth the precondtoner arsng from the second-order dscretzaton whch shows the scalablty wth the problem sze. Keywords: Posson equaton; cylndrcal coordnates; sphercal coordnates; symmetry constrant; Fast Fourer Transform; B-CGSTAB method 1 Introducton In many physcal problems, one often needs to solve the Posson equaton on a three-dmensonal non-cartesan doman, such as cylndrcal or sphercal domans. Correspondng author. Department of Appled Mathematcs, Natonal Chao Tung Unversty, 1001, Ta Hsueh Road, Hsnchu 300, TAIWAN. E-mal: mcla@math.nctu.edu.tw 1

2 For example, the projecton method n the smulaton of ncompressble flow n a ppe requres solvng the pressure Posson equaton. It s convenent to rewrte the equaton n those coordnates. The frst problem that must be dealt wth s the coordnate sngulartes (or poles) caused by the transformaton. It s mportant to note that the occurrence of those sngulartes s due to the representaton of the governng equaton n those coordnates and the soluton tself s regular f the source functon and the boundary condtons are smooth. For the past few years, the frst author and hs collaborators have developed a class of FFT-based fast drect solvers for posson equaton on 2D [1] and 3D [2] cylndrcal and sphercal domans. The methods have three major features, namely, the coordnate sngulartes can be treated easly, the resultng lnear equatons can be solved effcently by exstng avalable fast algorthms, and the dfferent boundary condtons can be handled wthout substantal dfferences. Besdes, the method s easy to mplement. Despte those aforementoned advantages of our algorthm, the numercal schemes n 3D doman [2] are only second-order accurate. Recently, the frst author has developed a smple compact fourth-order Posson solver on 2D polar geometry [3]. In fact, the scheme n [3] s formally fourth-order meanng that t has fourth-order accuracy only for the problem excludng the polar orgn but degrades to thrd-order accuracy when the orgn s ncluded. Despte the degradaton of one order of accuracy due to the presence of pole, the scheme handles the pole naturally; thus, no pole condton s needed. There are a few papers n the lterature that dscuss fourth-order fnte dfference schemes for the Posson equaton n 2D polar [6, 4, 5] and 3D cylndrcal coordnates [6, 7]. However, those papers need to derve some specal equatons at r = 0 (that s, pole condton). In ths artcle, we shall extend the methodology presented n [3] to the three-dmensonal cylndrcal and also sphercal geometres. 2 Posson equaton n 3D cylndrcal coordnates The Posson equaton n a fnte crcular cylnder Ω = {0 < r 1, 0 θ < 2π, 0 z 1} can be convenently represented n cylndrcal coordnates as 2 u r + 1 u 2 r r u r 2 θ + 2 u = f(r, z, θ), (2.1) 2 z2 u(r, 1, θ) = u T (r, θ), u(r, 0, θ) = u B (r, θ), u(1, z, θ) = u S (z, θ). (2.2) Here, we restrct the Drchlet boundary condtons on the top, bottom and the sdewall boundares. 2

3 2.1 Fourer mode equatons Snce the soluton u s perodc n θ, we can approxmate t by the truncated Fourer seres as u(r, z, θ) = N/2 1 n= N/2 where û n (r, z) s the complex Fourer coeffcent gven by û n (r, z) = 1 N N 1 k=0 û n (r, z) e nθ, (2.3) u(r, z, θ k ) e nθ k, (2.4) and θ k = 2kπ/N wth N the number of grd ponts along a crcle. The above transformaton between the physcal space and Fourer space can be effcently performed by the Fast Fourer Transform (FFT) wth O(N log 2 N) arthmetc operatons. Substtutng the expansons of (2.3) nto Eq.(2.1), and equatng the Fourer coeffcents, we derve û n (r, z) satsfyng the PDE 2 û n r r û n r + 2 û n z n2 2 r û 2 n = ˆf n (r, z), 0 < r 1, 0 z 1. (2.5) û n (r, 0) = û n B(r), û n (r, 1) = û n T (r), û n (1, z) = û n S(z). (2.6) Here, the nth Fourer coeffcent of the rght-hand sde functon ˆf n (r, z) and the boundary values û n S (z), ûn T (r), ûn B (r) are defned n a smlar fashon as (2.4). In the followng subsecton, we shall use the notatons U(r, z) = û n (r, z) and F (r, z) = ˆf n (r, z), respectvely. Usng the truncated Fourer seres expanson, the orgnal 3D Posson equaton (2.1) now becomes a set (N) of 2D Fourer mode equatons (2.5). In fact, we only need to solve half of Fourer modes, say n = 0, 1,..., N/2 1 snce u s a real valued functon and we have u n (r, z) = u n (r, z). Furthermore, snce those Fourer mode equatons are fully decoupled, they can be solved n parallel. After we solve those Fourer mode equatons and obtan the values of û n (r, z), the soluton u(r, z, θ) can be obtaned va the nverse FFT as (2.3). In [2], we have developed a second-order fnte dfference scheme to solve the Fourer mode equaton (2.5). In ths paper, our goal s to derve a formally fourth-order accurate compact scheme for the equaton (2.5). 2.2 Formally fourth-order compact dfference dscretzaton In order to derve a fourth-order fnte dfference approxmaton to Eq.(2.5), obvously, the frst and second dervatves, U r, U rr and U zz, must be approxmated to fourth-order accurately. To proceed, let us wrte down some dfference formulas for 3

4 the frst and second dervatves wth the truncaton errors O( r 4 ) and O( z 4 ) as follows. U r = δ 1 ru r2 6 U rrr + O( r 4 ), (2.7) U rr = δ 2 ru r2 12 U rrrr + O( r 4 ), (2.8) U zz = δ 2 zu z2 12 U zzzz + O( z 4 ). (2.9) Here δ 1 r, δ 2 r and δ 2 z are the centered dfference operators for the frst and second dervatves, defned as δru 1 j = U +1,j U 1,j, δ 2 2 r ru j = U +1,j 2U,j + U 1,j, (2.10) r 2 δ 2 zu j = U,j+1 2 U,j + U,j 1 z 2, (2.11) where U j are the dscrete values defned at the grd ponts (r, z j ). As n [2], we choose a shfted grd to avod the polar sngularty as r = ( 1/2) r, z j = j z, (2.12) for 1 L + 1; 0 j M + 1, wth r = 2/(2L + 1) and z = 1/(M + 1). Notce that, unlke the tradtonal mesh [9], we do not put the grd ponts on the polar axs drectly, thus; no pole condtons are needed. In order to have fourth-order approxmatons for U r, U rr and U zz, we need to approxmate the hgher order partal dervatves U rrr, U rrrr and U zzzz n Eqs.(2.7), (2.8) and (2.9) to be second-order accurate. In addton, those approxmatons should nvolve at most the neghborng nne-pont stencl to meet the compact requrement. To accomplsh ths, we dfferentate Eq.(2.5) wth respect to r and z and obtan the hgher order partal dervatves of U as U rrr = F r U rr r n2 U r 2 r 2n2 r U U zzr, (2.13) 3 U rrrr = F rr F r r n2 U r 2 rr 3 + 5n2 U r 3 r + 8n2 r U + U zzr 4 r U zzzz = F zz U rrzz U rzz r U zzrr, (2.14) + n2 r 2 U zz. (2.15) Now the partal dervatves U rrr, U rrrr and U zzzz are wrtten n terms of lower order partal dervatves whch are no hgher than second-order n r and z. Usng the standard centered dfference approxmatons to those lower order partal dervatves 4

5 Fgure 1: A nne-pont compact stencl. n Eqs.(2.13)-(2.15) and substtutng those approxmatons nto the equatons (2.7)- (2.9) and (2.5), we obtan the followng dfference scheme δru 2,j r2 12 [δ2 rf,j 1 δ r rf 1,j n2 δ r ru 2 2,j 3 + 5n2 δ r ru 1 3,j + 8n2 U r 4,j + 1 δ r rδ 1 zu 2,j δrδ 2 zu 2,j ] + 1 δ r ru 1,j r2 [δ 6r rf 1,j 1 δ r ru 2,j n2 r 2 δ 1 ru,j 2n2 r 3 U,j δrδ 1 zu 2,j ] n2 U r 2,j +δzu 2,j z2 12 [δ2 zf,j δzδ 2 ru 2,j 1 δ r zδ 2 ru 1,j + n2 δzu 2,j ] = F,j, (2.16) for 1 L, 1 j M. Note that, the scheme nvolves centered dfference approxmatons to frst or second order partal dervatves n r and z so only a nnepont compact stencl s used, see the Fgure 1 for llustraton. One can also see that f the order terms of r 2 and z 2 are gnored n the equaton (2.16), then the scheme recovers to the usual second-order accurate scheme as n [2]. In order to close the lnear system, the numercal boundary values U 0,j, U L+1,j and U,0, U,M+1 should be suppled. The numercal boundary value U 0,j can be gven by U 0,j = ( 1) n U 1,j due to the symmetry constrant of Fourer coeffcents (û n ( r/2, z j ) = ( 1) n û n ( r/2, z j )) [3]. And the other numercal boundary can be easly obtaned by the gven Drchlet boundary values U L+1,j = û n S (z j), U,0 = û n B (r ) 5 r 2

6 and U,M+1 = û n T (r ). 2.3 Numercal results In ths subsecton, we perform some numercal tests on the accuracy and effcency of our scheme. Snce the matrx of the resultant lnear system n (2.16) s nonsymmetrc, we use the BConjugate Gradent Stablzed method (B-CGSTAB) [10] to solve the lnear systems. The stoppng crteron of the convergence s based on the relatve resdual whch the tolerance ranges from dependng on the dfferent Fourer modes. Table 1 shows the maxmum relatve errors for three dfferent solutons of Posson equaton n cylndrcal coordnates. In all our tests, we use L mesh ponts n the radal (r) and axal (z) drectons, and 2L ponts n the azmuthal (θ) drecton. The rate of convergence s computed by the formula log 2 ( E L/2 E L ), where E L s the maxmum relatve error wth mesh resoluton L. From Table 1, we can see that the errors of the solutons show thrd-order convergence for all examples n the case of sold cylnder (0 < r 1). The loss of one order of accuracy seems to come from the dscretzaton near the polar orgn. Ths can be seen from the followng truncaton error analyss. In the Fourer mode equaton (2.5), the U r (= û n / r) term s dvded by r. So the second-order approxmaton of U rrr n (2.7) s dvded by an O( r) term near the orgn, whch makes the approxmaton of U rrr /r frst-order accurate. Ths has the consequence that the overall truncaton error of the U r /r term n the vcnty of the orgn s O( r 3 ) and thus so s the Fourer mode equaton (2.5). However, ths loss of accuracy does not appear when solvng the problem on a cylnder that excludes the polar sngularty such as the case of 0 < a r 1. Let us explan why s the case next. As n the sold cylnder case, we need to solve Eq.(2.5) wth the Drchlet boundary condton at r = 1 and an addtonal boundary condton mposed at r = a. Instead of settng a grd as n (2.12), we choose a regular grd n the radal drecton as r = a + r, = 0, 1,..., L, L + 1, (2.17) wth the mesh wdth r = (1 a)/(l + 1). Now the second-order approxmaton of U rrr n (2.7) s dvded by an O(a + r) term nstead of an O( r) term, so the truncaton error of the U rrr /r term s stll O( r 2 ) whch makes the dscretzaton error of U r /r term s O( r 4 ). Therefore, the overall truncaton error of Eq.(2.5) s O( r 4 ). One can see n Table 1 that the fourth-order convergence ndeed can be acheved for all test examples for the case of 0.5 r 1. In order to speed up the convergence of B-CGSTAB teraton, we have appled dfferent precondtoners whch nclude Block Jacob (BJ) [8], Incomplete LU factorzaton (ILU) [8], and the Fast Drect Solver (FDS) arsng from the second-order dscretzaton for the equaton (2.5) whch s developed n [2]. Here, we solve the 6

7 0 < r < r 1 L E L Rate E L Rate r cos θ+r sn θ+z u(r, z, θ) = e E E E E E E E E u(r, z, θ) = r 3 (cos θ + sn θ)z(1 z) E E E E E E E E u(r, z, θ) = cos(π(r 2 cos 2 θ + r sn θ)) sn(πz 2 ) E E E E E E E E Table 1: The maxmum relatve errors for dfferent solutons to Posson equaton n cylndrcal coordnates. dfference equaton (2.16) wth the Fourer mode number n = 1. The tolerance for the relatve resdual s chosen as Table 2 shows the number of teratons needed to be convergent for dfferent precondtoners appled to B-CGSTAB teraton. The column of B-CGSTAB s the one wthout any precondtoner whch, as expected, has the largest number of teratons. The precondtoners BJ and ILU both need double teratons when the grd ponts are doubled. One can see that, the FDS precondtoner turns out to be the most effcent one snce t has the least number of teratons, and the teratons are kept to be a constant when we double the grd ponts. L B-CGSTAB BJ ILU FDS Table 2: The performance comparson of dfferent precondtoners for the cylndrcal case. 7

8 3 Posson equaton n 3D sphercal coordnates In ths secton, we perform the smlar dervaton as the cylndrcal case and develop a spectral/fnte dfference scheme for Posson equaton n 3D sphercal doman. The Posson equaton wth Drchlet boundary n a sphercal shell Ω = {R 0 1, 0 φ π, 0 θ 2π} can be wrtten n sphercal coordnates as 2 u + 2 u u 2 φ + cot φ 2 2 u φ sn 2 φ 2 u = f(, φ, θ), (3.18) θ2 u(r 0, φ, θ) = u I (φ, θ), u(1, φ, θ) = u S (φ, θ). (3.19) Here, the boundary condton should be mposed on the nner ( = R 0 > 0) and outer ( = 1) surfaces of the sphere. As the cylndrcal case, the man dffculty for solvng Eq.(3.18) s to treat the coordnate sngulartes along the polar axs where north (φ = 0) and south (φ = π) poles are located. Most of numercal approaches ncludng fnte dfference and spectral methods nvolve mposng addtonal pole condtons to capture the behavor of the soluton n the vcnty of the poles. In [1, 2], we have developed a seres of FFT-based second-order fast Posson solver wthout pole condtons for 2D and 3D sphercal domans. In the followng, we wll develop a formally fourth-order compact scheme for Eqs. (3.18)-(3.19). To the best of our knowledge, we have not seen any related work n the lterature. 3.1 Fourer mode equatons As n the cylndrcal coordnate case, we approxmate u by the truncated Fourer seres as u(, φ, θ) = N/2 1 n= N/2 where û n (, φ) s the complex Fourer coeffcent gven by û n (, φ) = 1 N N 1 k=0 û n (, φ) e nθ, (3.20) u(, φ, θ k ) e nθ k, (3.21) and θ k = 2kπ/N and N s the number of grd ponts along a lattude crcle. Once agan, the above transformaton between the physcal space and Fourer space can be effcently performed by the Fast Fourer Transform (FFT) wth O(N log 2 N) arthmetc operatons. The expanson for the functon f can be wrtten n the smlar fashon. 8

9 Substtutng those expansons nto Eq.(3.18), and equatng the Fourer coeffcents, û n (, φ) then satsfes the PDE 2 û n û n û n 2 φ 2 + cot φ û n 2 φ n2 2 sn 2 φûn = ˆf n (, φ), (3.22) û n (R 0, φ) = û n I (φ), û n (1, φ) = û n S(φ). (3.23) Here, û n I (φ) and ûn S (φ) are the nth Fourer coeffcent of u I(φ, θ) and u S (φ, θ), respectvely. Next, we need to derve a formally fourth-order compact scheme for the Fourer mode equatons (3.22). 3.2 Formally fourth-order compact dfference dscretzaton As n [2], we choose a grd n (, φ) plane by = R 0 + r, φ j = (j 1/2) φ, (3.24) for 0 L + 1, 0 j M + 1 wth = (1 R 0 )/(L + 1) and φ = π/m. By the choce of those mesh ponts, we avod placng ponts drectly at north (φ = 0) and south (φ = π) poles. Agan, let the dscrete values be denoted by U(, φ j ) û n (, φ j ), and F (, φ j ) ˆf n (, φ j ). Our goal s to derve a fourth-order fnte dfference approxmaton to Eq.(3.22). Obvously, the frst and second dervatves, U, U, U φ and U φφ must be approxmated to fourth-order accurately. In order to have fourth-order approxmatons for U, U, U φ and U φφ, we need to approxmate the hgher order dervatves U, U, U φφφ and U φφφφ to be second-order accurate. We then reduce those hgher order partal dervatves to lower order by dfferentatng the equaton (3.22) and use the regular centered dfference to approxmate those lower order partal dervatves. The dervaton s very smlar to the cylndrcal case, so we omt the detal here. After a tedous calculaton, we obtan a fnte dfference scheme as follows. 9

10 For 1 L, 1 j M, we have δu 2,j 2 12 [δ2 F,j 2 δ F 1,j + ( 8 + n2 csc 2 φ j +( 8 6n2 csc 2 φ j )δ 1 U,j δ 2 φu,j 10 cot φ j 4 )δu 2,j + 10n2 csc 2 φ j U 4,j δφu 1,j + 6 cot φ j δ δ 1 φu 1 3,j δδ 1 φu 2,j cot φ j δ δ 2 φu 1 2,j 1 δ δ 2 φu 2 2,j ] + 2 {δ U 1,j r2 6 [δ1 F,j 2 δ 2 U,j + ( 2 + n2 csc 2 φ j 2 )δu 1,j + 2 cot φ j δ φu 1 3,j δφu 2,j cot φ j δ δ 1 φu 1 2,j 1 δ δ 1 φu 2 2,j 2n2 csc 2 φ j U 3,j ]} + 1 {δ φu 2 2,j φ2 12 [2 δφf 2,j 2 cot φ j δφf 1,j +((3 + n 2 ) csc 2 φ j 1)δφU 2,j + ( 3 5n 2 ) csc 2 φ j cot φ j δφu 1,j +2n 2 csc 2 φ j (4 csc 2 φ j 3)U,j + 2 cot φ j δ 1 φδ 1 U,j + 2 cot φ j δ 1 φδ 2 U,j 2 δφδ 2 U 1,j 2 δφδ 2 U 2,j ]} + cot φ j {δ φu 1 2,j φ2 6 [ 2 δφδ 1 U 2,j 2 δφδ 1 U 1,j +(1 + n 2 ) csc 2 φ j δφu 1,j cot φ j δφu 2,j 2n 2 csc 2 φ j cot φ j U,j + 2 δφf 1,j ]} n2 csc 2 φ j U 2,j = F,j. (3.25) Note that, the scheme nvolves centered dfference approxmatons to frst or second order partal dervatves n and φ so only a nne-pont compact stencl s used. One can also see that f the order terms of 2 and φ 2 are gnored n the equaton (3.25), then the scheme recovers to the usual second-order accurate scheme as n [2]. When j = 1 for Eq.(3.25), the numercal boundary value U,0 can be gven by U,0 = ( 1) n U,1. Ths s because the Fourer coeffcent satsfes the symmetry constrant û n (, φ/2) = ( 1) n û n (, φ/2) [2]. Smlarly, another numercal boundary value U,M+1 can also be obtaned by U,M+1 = ( 1) n U,M for the same reason. So the numercal boundary values n the φ drecton are provded and no pole condton s needed n our fnte dfference settng. The other numercal boundary U 0,j, U L+1,j are gven by the boundary values û n I (φ j), û n S (φ j). 3.3 Numercal results In ths subsecton, we perform smlar numercal tests on the accuracy and effcency of our scheme for the sphercal coordnates case. In all our tests, we use L mesh ponts n the radal () and colattude (φ) drectons, and 2L ponts n the longtude (θ) drecton. The nner radus s chosen as R 0 = 0.5. The dfference equaton (3.25) s solved by the B-CGSTAB method for n = 0, 1,... L 1 where the tolerance of 10

11 stoppng ranges from dependng on the dfferent Fourer modes. Once we obtan the Fourer coeffcents, the numercal soluton can be computed by the nverse FFT as n (3.20). Table 3 shows the maxmum errors of the method for three dfferent solutons of Posson equaton n a sphercal shell. One can see that the errors of the solutons show thrd-order convergence for all test examples. The reason of losng one order of accuracy s exactly the same as the cylndrcal case so we omt the dscusson here. Table 4 shows the number of teratons needed for solvng the dfference equaton (3.25) of n = 1 by the B-CGSTAB method wth dfferent precondtoners. The tolerance of stoppng for the relatve resdual s chosen as Generally speakng, the performance of those precondtoners are almost the same as the cylndrcal case. It s nterestng to see that the ILU precondtoner seems to have the less number of teratons than the FDS when the number of grd ponts s small. However, the ILU does ncrease the teratons as the grd ponts ncreasng whle the FDS precondtoner stll keeps the number of teratons. L E L Rate sn φ cos θ+ sn φ sn θ+ cos φ u(, φ, θ) = e E E E E u(, φ, θ) = 3 (cos θ + sn θ) sn φ(1 cos φ) E E E E u(, φ, θ) = cos(π( 2 cos 2 θ sn 2 φ + sn θ sn φ)) sn(π 2 cos 2 φ) E E E E Table 3: The maxmum relatve errors for dfferent solutons to Posson equaton n sphercal coordnates. 4 Concluson and acknowledgement In ths paper, we present a formally fourth-order compact dfference scheme for 3D Posson equaton n cylndrcal and sphercal coordnates. The solver reles on the 11

12 L B-CGSTAB BJ ILU FDS Table 4: The performance comparson of dfferent precondtoners for the sphercal case. truncated Fourer seres expanson, where the partal dfferental equatons of Fourer coeffcents are solved by a formal fourth-order compact dfference dscretzatons wthout pole condtons. The resultant lnear system s then solved by the B- CGSTAB method wth dfferent precondtoners. The numercal results confrm the formal accuracy of our scheme. Meanwhle, the precondtoner arsng from the second-order fast drect solver shows the scalablty of B-CGSTAB wth the problem sze. The authors are supported n part by the Natonal Scence councl of Tawan under research grant NSC M References [1] M.-C. La and W.-C. Wang, Fast drect solvers for Posson equaton on 2D polar and sphercal gepmetres. Numer. Meth. Partal Dff. Eq., 18, 56-68, (2002). [2] M.-C. La, W.-W. Ln and W. Wang, A fast spectral/dfference method wthout pole condtons for Posson-type equatons n cylndrcal and sphercal geometres. IMA J. Numer. Anal., 22, , (2002). [3] M.-C. La, A smple compact fourth-order Posson solver on polar geometry. J. Comput. Phys., 182, , (2002). [4] R. C. Mttal and S. Gahlaut, Hgh-order fnte-dfference schemes to solve Posson s equaton n polar coordnates, IMA J. Numer. Anal., 11, 261, (1991). [5] M. K. Jan, R. K. Jan and M. Krshna, A fourth-order dfference scheme for quaslnear Posson equaton n polar co-ordnates, Communcatons n Numercal Methods n Engneerng, vol. 10, (1994). [6] S. R. K. Iyengar and R. Manohar, Hgh order dfference methods for heat equaton n polar cylndrcal coordnates, J. Comput. Phys., 77, 425 (1988). 12

13 [7] S. R. K. Iyengar and A. Goyal, A note on multgrd for the three-dmensonal Posson equaton n cylndrcal coordnates, J. Comput. Appl. Math., 33, , (1990). [8] B. Rchard et al. Templates for the soluton of lnear systems:buldng blocks for teratve methods, SIAM (1994). [9] J. C. Strkwerda, Fnte Dfference schemes and Partal Dfferental Equatons, Page , Wadsworth & Brooks/Cole, (1989). [10] H. A. Van der Vorst, B-CGSTAB : A fast and smoothly convergng varant of B-CG for the soluton of nonsymmetrc lnear systems. SIAM J. Sc. Statst. Comput., 13, , (1992). 13