ORNL/CP-96473

IMPOSSIBILITY OF UNCONDITIONAL STABILITY AND ROBUSTNESS OF DIFFUSIVE ACCELERATION SCHEMES

Y. Y. Azmy Oak Ridge National Laboratory*

Oak Ridge, Tennessee 3783 1-6363

yya@ ornl.gov

P.O. BOX 2008, MS-6363

Phone: (423) 574-8069 Fax: (423) 574-9619

"The submitted manuscript has been authored by a contractor of the US. Government under contract DE-AC05-960R22464. Accordingly, the US. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for US. Government purposes."

19980406 138 Paper submitted for publication and oral presentation, I998 American Nuclear Society Radiation Protection and Shielding Division Topical Conference, Nashville, TN, April 19-23, 1998.

*Research sponsored by the Oak Ridge National Laboratory managed by Lockheed Martin Energy Research Corporation for the U.S. Department of Energy under contract No. DE-AC05-960R22464.

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use- fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spe- cific commercial product, process, or service by trade name, trademark, manufac- turer, or otherwise does not necessarily constitute or imply its endorsement, rccom- mendktion, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Impossibility of Unconditional Stability and Robustness of Diffusive Acceleration Schemes

Y. Y. Azmy

Abstract

We construct a problem for which exists no preconditioner with a cell-centered diffusion coupling stencil that is unconditionally stable and robust: In particular we consider an asymptotic limit of the Periodic Horizontal Interface (PHI) configuration wherein the cell height in both layers approaches zero like u2 while the total cross section varies like B in one layer and like l/u in the other layer. In such case we show that the conditions for stability and robustness of the flat eigenmodes of the iteration residual imply instability of the modes flat in the y-dimension and rapidly varying in the x-dimension.

1. INTRODUCTION

Recently the Adjacent-cell Preconditioner (AP) scheme was devised to address some of the limitations of DSA'-3 in multidimensional ge~rnetry.~~' Namely, AP applies a cell-centered diffusion coupling stencil thus comprising a smaller algebraic system than edge-centered DSA, and achieves better spectral properties in model problem configurations? By definition AP includes cell-centered DSA schemes as special cases, but it is not constrained in the choice of the diffusion parameters except to require stability of the accelerated iterations. Spectral analysis of AP predicted its unconditional stability and robustness, i.e. spectral radius boundedfar below one, for model problems over a wide range of problem parameters particularly cell size and total cross s e ~ t i o n . ~ , ~ However, in multidimensional non-model problems AP with a reciprocal-averaging mixing formula suffered significant inefficiency with increasing discontinuity in the total cross section.'

Horizontal Interface (PHI) configuration with no success,6 and the question became whether there is any preconditioner with a diffusion stencil that possesses this property. In this paper we show that this is not possible. Our conclusion extends beyond AP to include all cell-centered diffusion operators, and based on numerical experiments we conjecture it will also hold for edge-centered schemes. Also while the analysis presented here is performed for the Zero-order Nodal Integral Method (ONIM)' which continuously spans the entire range of cell sizes with the correct limits of the spatial weights, our conclusion is valid for all spatial discretizations of S, methods that possess correct limits of the spatial weights, namely Diamond Difference (DD), and Step for thin, and thick cells, respectively.

Many attempts were made to find a better, unconditionally stable mixing formula for the Periodic

2. STABILITY CONDITIONS FOR THE FLAT EIGENMODES OF PHI

The PHI configuration was described in Ref. 6 together with the equations describing the spectral behavior of the iteration residual for the Source Iterat'ons (S Fourier modes of the previous iteration residual 6' = 15; , $:! are mapped into a'+' according to

and the AP schemzs. For the latter,

-t 6'+' = M(r,s) 6' = [B +D-' (B -I)1ap , (1)

where r, s are the Fourier variables; B is the SI mapping of the iteration residual; D is the Fourier- decomposed AP operator; and I is the identity matrix.

the flat eigenmode, r, s - 0. In this limit, We now study the behavior of M near the origin in Fourier space and derive stability conditions for

(2 ) B - Bo + r2BrZ + s2Bs2

where the first order terms vanish due to the symmetry of the angular quadrature, and

The elements Ej, p J , j / , j = 1,2, and t = r or s, are obtained by taking the appropriate limits of the full expression for B.6 It is easy to show that in case of perfect scattering, C = I if the preconditioner is not singular at r = s = 0 then the spectral radius of M is bounded from below by 1, implying lack of stability. Hence a necessary condition for stability of the preconditioned iterations is the vanishing of the determinant of D at the origin in Fourier space

’

5,Ez =4D,.,, D,,, , Ej 5 Dj + 2 D x , j , j = 1,2 . (4 )

Of course in this case M does not exist at the origin, and only its limit as r, s - 0 can be considered. After some algebra the limit of the elements of M are given by

M. . E - Ejfij I + - 2 Ej Dyj + ( 1 - E j ) + z [ ( 5 j / p J j 1 + 4 E j D x j l - 2 D , j ~ ~ j l ) r z + ( b j l p ; j + E I ’ D y j - 2 D y j p ; j , ) s 2 ] , (5.a) A I . I

and the determinant of the preconditioner at the origin, in view of Eq. (4), becomes

(6) - A 5 - 4 ( b 1 D , , + b , D x 1 ) r 2 + 4 D , , D , , s 2 .

It is evident that M has two scales near the origin in Fourier space, the first diverges like 0( i2 , s-’), and the other is O( 1). Separating the divergent component then requiring the O(r-’, s-’) term in the asymptotic expansion of the spectrum to vanish exactly yields the condition

E l b 2 + E z b I + 2 ( E 2 D Y 1 + E , D y 2 ) = 0 . (7)

Computing the eigenvalues of M under the condition, Eq. (7), yields an expression that diverges at the origin as O(r,s) unless the First Stability Condition (ISC)

b i = D i + 2 D x , i = - 2 D y , i , i = 1 , 2 , (8)

D i = 2 ( E i / E i l ) D , , i I , i ’ ~ i = 1,2 . (9)

or the Second Stability Condition (2SC)

Satisfaction of at least one of the two conditions ISC and 2SC is necessary for the flat mode to be stable.

3. IMPOSSIBILITY OF UNCONDITIONAL STABILITY AND ROBUSTNESS

0 -2pa- -+12p2 a2

In order to establish the impossibility of unconditional stability of any AP type acceleration scheme we need only construct one case where no choice of the preconditioner parameters results in a spectral radius smaller than 1. This is accomplished for the specific PHI configuration: uI = l/a, = u , b, = b, = 02,

in the limit o+O. First we determine the asymptotic behavior of the various coefficients in the B matrix in the same a-limit first near the origin

1 - 2 -P 12

1 7 -P 12 I

where the super bar indicates a weighted sum over discrete ordinates, e.g., F=Cp,rlop, and we have assumed perfect scattering, cI = c, = 1. These expressions define the asymptotic behavior of the SI scheme near the origin in Fourier space for the case under consideration when a-0.

Next we evaluate the B matrix at the eigenmode r=n/2, s=Oin the limit u - 0 starting with the expressions for the A matrix derived earlier

0 B(d2,O) =

0

(lO.a)

(lO.b)

(1O.c)

3.1 THE FIRST STABILITY CONDITION

Now we apply ISC to AP and show that the robustness of the flat mode, r, s -+ 0 results in an unbounded eigenvalue at r = d 2 , s = 0.

3.1.1 Robustness of the flat eigenmode: r, s -+ 0

Rewriting the limit of the eigenvalues of the M matrix under the First Stability Condition at the origin in Fourier space

(1 2.a)

Now we compute the leading term of each eigenvalue along the r- and s-axes keeping in mind that the asymptotic behavior of the preconditioner parameters with respect to u is yet undetermined. Starting with v- along the r-axis, and allowing the O(a-’) terms to dominate,

which diverges as u+O unless Dy,j = O(uyy+j), j = 1,2, and Ny,j s -2. Furthermore, in order to prevent a unity eigenvalue (which would imply lack of robustness) it is necessary that

Ma[! , , , , , Ny,2] = - 2 ( 1 4 )

Next consider v- along the s-axis

which implies that a necessary condition for robustness is

We now use Eqs. (14) and (16) to show that DyJ dominates D, fo r j = 1,2, as u + 0. First we note that Eq. (16) amounts to

Min[N,,, + N V , l , N x , l + N y , 2 ] = Min[N,, , + 1,Ny,2 + 1 1 (17 )

where N x j are defined in analogy to NYj. Suppose by Eq. (14) Ny,j s Ny,2 = -2; then either

N x , 2 + N y , I ~ N x . 1 + N y , 2 = N y . I * 1 = T , 2 2 1 ’ N y , 2 and NX.1 = N y , 1 + 3 > N , : t or

N,, + Ny,2 2 Nx,2 + Ny. = N,. + 1 ==) N,. 2 Ny. I + 3 > Ny. 1 and Nx,2 = 1 > 4 . 2

which amount to N x j > N y , j , j = 1,2 ,

Similarly, if by Eq. (14) 5 = -2, the same Inequality (18) is reached. We now use this inequality to show that the eigenvalue at r = 7d2, s = 0, is O(U-’).

3.1.2 Instability of the Eigenmode r = 762, s = 0

Upon applying ISC the inverse of the preconditioner matrix at this eigenmode we obtain

This can be simplified in the limit u - 0 by using Inequality (18) in the numerators of the diagonal elements, and by noting that the denominator can be rewritten as

The preconditioned iterations for this eigenmode, upon applying lSC, are governed by

Clearly the two rows are identical, from which it follows that a lower bound on the spectral radius is provided by

The last term in the expression for pPb is O ( d ) by virtue of Eq. (16) implying instability in the limit a-0.

3.2 THE SECOND STABILITY CONDITION

In this section we apply 2SC to AP and show that a necessary condition for the robustness of the flat mode, r, s-+O results in an unbounded eigenvalue at r = n/2, s = 0.

3.2.1 Robustness of the Flat Eigenmode: r, s -+ 0

Unlike lSC, 2SC does not produce a simple discriminant for the expressions of v, in the neighborhood of the origin in Fourier space. However, it is possible to obtain simple formulas for the eigenvalues along the s-axis as follows. Using the asymptotic expansions of the Ej and pf JJ ., in the expressions for Moj,j,, j , j ' = 1,2, and t = r, s, then applying 2SC along the s-axis we obtain the two eigenvalues

(24.a)

(24.b)

( N,. j ) As in the case of lSC, here also v- diverges as u -+ 0 unless D YP J . = O u . , j = 1 ,2 and I -4, Ny.,2 I 0. Furthermore, in order to prevent a unity eigenvalue (which would imply lack of N

ro&stness) it is necessary that at least one equality holds, Le.

Inequality (25) automatically guarantees that v + = Ob') . Even though a simple form of the eigenvalues along the r-axis is not available, we derive conditions amounting to the dominance of the DYj

coefficients over the Dx,j / that are necessary to show that the eigenvalues at r = d2, s = 0, are unbounded. This is accomplished by substituting s = 0 in the formula for the eigenvalues of M, then after some simplifications

where we'have defined

1.

We consider two possibilities:

[ :) dominates [ q) as a-0: Then simplifying Eq. (26) we obtain

, so that robustness of this eigenmode requires

- CIaDy2 a2 2. [ pa7 %] dominates [ 7 as u -+ 0: Then simplifying Eq. (26) we obtain:

or

, D v- - - 4 p a7 2 , so that the stability of this eigenmode requires

U2 h

If neither term dominates the other we use either Eq. (28) or (29) since only the asymptotic order is concerned in the ensuing analysis. In the remainder of this subsection we use Eqs. (28) and (29) to derive conditions that will render the r = d 2 , s = 0 eigenmode highly unstable. Considering the asymptotic order in Eq. (28) we conclude that

Min[Nx, , + Ny,l + 4 , Nx,2 + Ny,2] = Ny,2 + 1 - If according to the Inequality (25) N, , = 0, N , , s -4, then either:

N , , , = l > N , , , , + 2 , and Nxsl + N , , l + 4 ~ N x ~ 2 ~ N x ~ l ~ -3-Ny,1>Ny,2

or

If according to Inequality (25) Ny,, = -4, Ny,2 r; 0, then either:

N,,, +& +4 = N y , 2 + 1 +,.1 L Ny,2 +1> N,,2 , and Nx,2 L 1 > Ny, l + 2 .

Now considering Eq. (29), we conclude that

Min[Nx. , +N, , , , +47 4 2 + N , , , ] = N y , , + 5

If according to Inequality (25) N , , = 0, Ny,, s -4,.then either:

Nxvl + NYsl + 4 = N , . , + 5 - NX,, = 1 >Ny,2 and Nx,2 2 N , , l + 5 > Nyn , + 2 , or

N x , 2 = N , , I +5>N,?*, + 2 , and N x , 1 + N , , 1 + 4 ~ N y , 1 + 5 - N x . I ~ 1>Ny ,2

Hence, also Eq. (29) implies Inequality (31). Now we use Inequality (31) to show that the eigenvalue at r = nI2, s = 0 is unbounded.

3.2.2 Instability of the Eigenxnode r = 75'2, s = 0

Upon applying 2SC to the inverse of the preconditioner matrix at this eigenmode, then taking the limit (J + 0 and using Inequality (3 1) we obtain

(33) 1 o4Dqy,1 a2Dy.I

D-' ( d 2 , O ) = - 4(a4 D ~ . ~ D , , i + D ~ , 2 D y , 2 ) [ O2 Dy,2 q y . 2 ]

The preconditioned iterations for this eigenmode, upon applying 2SC, are governed by

The supremum of the magnitude of the two eigenvalues of the matrix M( d2,O) is bounded from below by half the sum of its diagonal elements which can be simplified to

The first term in the expression for ptb is O ( d ) by virtue of Eqs. (28) and (29) implying instability in the limit a+O.

5. CONCLUSION

We have shown that there exists no preconditioner with the Ap cell-centered coupling stencil that is unconditionally stable and robust. The only conditions applied to derive this conclusion for the ONIM are the two Stability Conditions (one at a time) and the robustness of the flat mode. If the latter condition is not imposed, i.e., the eigenvalue near the origin in Fourier space is allowed to approach unity as a+O, as indeed is the case with the reciprocal-averaging mixing formula, then the r = d 2 , s = 0, mode's eigenvalue remains finite. Hence, we conclude that in the context of the PHI analysis reciprocal- averaging mixing provides as small a lower bound on the spectral radius, 1, as can be achieved in the limit a+O. To achieve unconditional stability and robustness it is necessary to consider novel preconditioners whose coupling stencils might be more complex than the diffusion operator.

, I . .

ACKNOWLEDGEMENT

This research was sponsored by the Oak Ridge National Laboratory managed by Lockheed Martin Energy Research Corporation for the U.S. Department of Energy under contract No. DE-ACOS- 960R22464.

REFERENCES

1. W. H. Reed, “The Effectiveness of Acceleration Techniques for Iterative Methods in Transport Theory,” Nucl. Sei. Eng., 45, 245 (1971).

2. R. E. Alcouffe, “Diffusion Synthetic Acceleration Methods for the Diamond-Differenced Discrete Ordinates Equations,” Nucl. Sei. Eng., 64, 344 (1977).

3. E. W. Larsen, “Unconditionally Stable Diffusion Synthetic Acceleration Methods for the Slab Geometry Discrete Ordinates Equations. Part I: Theory,” Nucl. Sei. Eng., 82,47 (1982).

4. Y. Y. Azmy, “Adjacent-cell Preconditioners for Solving Optically Thick Neutron Transport Problems,” in Proc. Eighth Intl. Con$ on Radiation Shielding, Arlington, Texas, April 24-27, 1994, Vol. I, p. 193, American Nuclear Society, La Grange Park, IL (1994).

5. Y. Y. Azmy, “Adjacent-cell Preconditioners for Accelerating Multidimensional Neutron Transport Methods,” in Proc. Eighth Topical Mtg. on Advances and Applications in Radiation Protection and Shielding, N. Falmouth, MA, April 21-25, 1996, Vol. 1, p. 390, American Nuclear Society, La Grange Park, IL (1996).

6. Y. Y. Azmy, “Analysis and Performance of Adjacent-cell Preconditioners for Accelerating Multidimensional Transport Calculations,” in OECD/NEA Mtg. on 3 0 Deterministic Radiation Transport Computer Programs, Features, Applications, and Perspectives, December 2-3, 1996, Paris, France, to be published.

7. Y. Y. Azmy, “The Weighted Diamond Difference Form of Nodal Transport Methods,” Nucl. Sci. Eng., 98,29 (1988).

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