Richardson Extrapolation-Based Verification Method of Scientific Calculation Program without the Oracles: A Case Studydownloads.hindawi.com/journals/mpe/2019/5782146.pdf ·...

Research ArticleRichardson Extrapolation-Based VerificationMethod of Scientific Calculation Program withoutthe Oracles A Case Study

Shiyu Yan12 Xiaohua Yang12 Guodong Cheng12 and Hua Liu 2

1Computer School University of South China Hengyang Hunan Province 421001 China2CNNC Key Laboratory on High Trusted Computing University of South China Hengyang Hunan Province 421001 China

Correspondence should be addressed to Hua Liu lhsmileusceducn

Received 1 October 2018 Accepted 2 December 2018 Published 30 January 2019

Academic Editor Fausto Arpino

Copyright copy 2019 ShiyuYan et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For the verification test of some scientific calculation programs different comparison methods are commonly applied to ensure thecorrectness of the computations However it is difficult to verify whether the testing output is correct because the oracles whichinclude the expected output are not always available or too hard to get For this reason the authors focus on using the RichardsonExtrapolation to estimate the convergences of the numerical solution on different levels of mesh refinement These numericalconvergence properties can be applied to verification test without the need for giving the oracles In the present study the authorstake the program test of themultigroup neutron diffusion equations as a study case andpropose the Richardson Extrapolation-basedverification method Three verification criterions are obtained based on our approach In addition a test experiment is conducteddemonstrating the validity of our theoretical results

1 Introduction

The importance of scientific software in mission-critical andsafety-critical applications has increased dramatically duringthe last three decades The engineers developers and usersof scientific software face a critical problem How shouldconfidence in computational science and engineering becritically assessed [1] Since any subtle defects in criticalsystem software will cause catastrophic consequences thecredibility of the computational results must be raised to ahigher level

Verification test is the major process for improving thisconfidence of the software quality Great progress has beenmade in the aspects of verification in scientific software [2]There remains a great challenge in this verification commu-nity that is there is no reliable test oracle to indicate whatthe correct output should be for arbitrary input which wascalled oracle problem This problem commonly arises whenconducting verification testing of scientific software manyscientific applications fall into the category of nontestable

programs [3] where an oracle is unavailable or too difficultto implement However many testers of scientific softwaregenerally adopt the following mechanisms as test oracles(a) comparing with analytical solutions simulation resultstabulated values or hand-calculations [4] (b) verifying withstandard mathematical libraries or reference software pack-ages [5] Additionally other methods for solving the oracleproblem have been developed including formal proof [6]analytic solutions for simplified physics method of manu-factured solutions [7] PDE benchmark solutions and meta-morphic testing [8 9] For example around 30 verificationbenchmarks have been developed by the National Agency forFinite Element Methods and Standards (NAFEMS) whichhave been used widely in verification In addition ANSYS[10] and ABAQUS have roughly 270 formal verificationtests Readers are referred to [11ndash13] for more details onverification The commonality of these methods is to use theanalytical solution simulation results and reference values astest oracle However these methods are unavailable for everynewly investigated program

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 5782146 9 pageshttpsdoiorg10115520195782146

2 Mathematical Problems in Engineering

In order to solve the oracle problem in scientific appli-cations the mathematical models and numerical methodsare factors that should be investigated again The a posterioriestimation methods based on Richardson Extrapolation areexamined [15] Emphasis is placed on discretization errorestimation methods based on Richardson Extrapolation [16]In the process of using Richardson Extrapolation to analyzethe order of error accuracy of the numerical solution theexact solution of the partial differential equation is actually anauxiliary role and does not need to be given In other wordsRichard Extrapolation methods can be applied to verify theno-test oracle problem Moreover the relevant verificationcriterions could be constructed by the analysis process basedon Richardson Extrapolation method [16]

Since the quality of the nuclear reactor design softwareis closely related to the safety and economy of reactorsthe verification test is very important for improving thecredibility of programs Actually the program for solvingmultigroup diffusion equations is the primary procedure ofthe nuclear reactor design software [14] In the critical stateof the reactor the multigroup diffusion equations are usuallydescribed as the solution of Lambda Modes problem or120582-eigenvalue problem [17] which provides the fundamen-tal eigenvalue (the maximal eigenvalue) that is called theeffective multiplication factor (k-effective) In this case theneutron flux is a corresponding eigenfunction Computationsof the Lambda Modes problem provide the results of thek-effective and neutron flux distribution for reactor designcalculations Hence if the program is calculated incorrectly itwill lead to error prediction and affect the safety and economyof the reactor Though traditional comparison methods arecommonly applied in verification test of partial differentialequations program it is difficult to give expected values toverify whether the calculation results are correct

In the present study the program of the multigroupneutron diffusion equations is performed as a study case Weinvestigate the mathematical models and the finite differencenumerical algorithm of the multigroup diffusion equationsBased on Richardson Extrapolation the convergence prop-erties of numerical solution in 1198670 and 1198671 are obtainedand then the rigorous verification criterions are constructedfor verification However we would not need to give exactsolutions to verification test

The rest of the paper is organized as follows In Section 2we introduce the mathematical model Section 3 presents themain numerical method and convergence analysis processbased Richardson Extrapolation Test experiments are con-ducted and the results are reported in Section 4 Finally wegive a summary and conclude the paper in Section 5

2 Background

21 Mathematical Model The steady state multigroup neu-tron diffusion equations are the basic mathematical model inreactor design where the governing equations are given by(1) and (2)

minus nabla sdot 119863119892 (119903) nabla120601119892 (119903) + Σ119877119892 (119903) 120601119892 (119903)

= 119866sum1198921015840=11198921015840 =119892

Σ1198921015840997888rarr119892 (119903) 1206011198921015840 (119903) + 120594119892119896119890119891119891119876(119903)

119892 = 1 2 119866(1)

119876 (119903) = 119866sum1198921015840=1

(120592Σ119891 (119903))1198921015840 1206011198921015840 (119903) (2)

where 119892 is the energy group119863119892(119903) is the diffusion coefficientof the energy group 119892 Σ119877119892(119903) is the removal cross section ofthe energy group 119892 Σ1198921015840997888rarr119892(119903) is the scattering cross sectionfrom energy group 1198921015840 to 119892 120592119892 is the average number ofneutrons emitted by fission of the energy group 119892 (sum119891(119903))1198921015840is the fission cross section of the energy group 1198921015840 120601119892(119903)is the neutron scalar flux of the energy group 119892 120594119892 is theintegrated fission spectrum of the energy group 119892 and 119896119890119891119891 isthe effectivemultiplication factor of reactor119876(119903) is the fissionterm In the multigroup formulation the neutron diffusionequations are represented by a coupled system of partialdifferential equations for the flux

Assuming a bounded 2D or 3D domain Ω with a convexboundary 120597Ω the domainΩ is composed of the finite numberof subassemblies Ω119894 119894 = 1 2 119872 For 119903 isin Ω with an outerboundary conditions we have

119863119892 120597120601119892 (119903)120597119899 + 120601119892 (119903) = 0 119903 isin 120597Ω (3)

where 119899 is the outer normal to the boundary 120597Ω at aninterface between dissimilar regions Ω119894 with continuousconditions we have

120601119892 (119903+) = 120601119892 (119903minus) (4)

and

119863119892 120597120601119892 (119903+)120597119899 = 119863119892 120597120601119892 (119903minus)120597119899 119903 isin 120597Ω119894 (5)

where 119903+ and 119903minus represent the value of both sides of the inter-face respectivelyThe quantitiesΣ119877119892 (119903)119863119892(119903)Σ1198921015840997888rarr119892(119903)120594119892120592Σ119891(119903) are constant in each Ω119894

To simplify the manipulations (1) and (2) are expressedin a matrix system by the following form

M120593 = 1119896119890119891119891F120593 (6)

where the vector of the neutron flux is 120593 = [1206011 1206012 120601119866]119879the coefficient matrixM = (119886119894119895)119866119866 is defined with

119886119894119895 = minus119863119894nabla2 + Σ119877119894 119894 = 119895minusΣ1198951015840119894 119894 = 1198951015840 (7)

and the coefficient matrix F = (119891119894119895)119866119866 is defined with 119891119894119895 =120594119894120592119895Σ119891119895

Mathematical Problems in Engineering 3

22 Power Iteration The power iteration (PI) method is themost used method to obtain the fundamental eigenvalue andits eigenvector [18]This iterative source method is describedin detail by [19] The power iteration algorithm process is asfollows

(I) Starting with a positive initial estimate for 120593(0) and119896(0)119890119891119891

they are substituted into the right hand of (6)to obtain the neutron flux 120593(1) The upper labelsrepresent the number of iterations

(II) For the i-th iteration (6) is written as

M120593(119894+1) = 1119896(119894)119890119891119891

S(119894) (8)

where S = F120593(III) Updating the value for new 119896119890119891119891 and the new flux vec-

tor 120593 119896(119894+1)119890119891119891 and S(119894+1) are calculated by the followingequations respectively

S(119894+1) = F120593(119894+1)119896(119894+1)119890119891119891 = 119896(119894)119890119891119891intΩ S

(119894+1)119889ΩintΩS(119894)119889Ω (9)

with the stop criterion as follows100381610038161003816100381610038161003816119896(119894+1)119890119891119891 minus 119896(119894)

119890119891119891

100381610038161003816100381610038161003816100381610038161003816100381610038161003816119896(119894+1)119890119891119891100381610038161003816100381610038161003816

lt 120576110038161003816100381610038161003816intΩ S(119894+1)119889Ω minus int

ΩS(119894)119889Ω1003816100381610038161003816100381610038161003816100381610038161003816intΩ S(119894+1)119889Ω10038161003816100381610038161003816

lt 1205762(10)

where 1205761 and 1205762 are small constants(IV) If the result does not correspond to the stop criterion

return to step (II)

The iteration process of step (II) is called inner iterationwhich usually applies the finite differences finite elementsand other methods to solve (1) Readers are referred to [17 2021] for more details on themathematical theory In particularthe finite difference method is a widely used method

At each step of the power iteration procedure if we omitthe superscript and subscript (1) is expressed as G uncoupledself-consistency elliptic boundary value problems Equation(1) is rewritten by

minusnabla119863 (119903) nabla120601 (119903) + Σ119877 (119903) 120601 (119903) = 119878 (119903) 119903 isin Ω (11)

119863(119903) 120597120601 (119903)120597119899 + 120601 (119903) = 0 119903 isin 120597Ω (12)

Here 119878(119903) contains the fission and scattering termsActually the fission source is not known so that the inneriterationmust be embedded in step (III) which is called outeriteration Therefore we fill the values of the inner iteration tothe outer iteration

i+1

2i Ximinus

1

2iminus1 i+10

middot middot middot middot middot middotmiddot middot middot middot middot middot

Figure 1 Space discrete in one-dimensional X geometry

3 Scientific Software and Numerical Method

Many large and complex reactor physics calculation softwarepackages include programs for solving the multigroup neu-tron diffusion equations which widely adopted the differ-ential method to carry on spatial discretization [22] In thecurrent study we design and compile a calculation programcalled HARMONY-2 which also adopts the finite differencemethod to discrete the multigroup neutron diffusion equa-tion and invokes ARPACK which is an open source packagedeveloped by the Department of Computing and AppliedMathematics of RICE University The ARPACK implementsthe IRAM (Implicitly Restarted Arnoldi Method) algorithmwhich is presented in [23] HARMONY-2 is used to calculatehigh order harmonics of the neutron diffusion equations

31 The Finite Difference Method We consider an exampleof one-dimensional and multigroup neutron diffusion equa-tions

minus 119889119889119903 (119863119892 (119903) 119889120601119892

119889119903 ) + Σ119877119892 (119903) 120601119892 (119903) = 119878119892 (119903) 119903 isin Ω 119892 = 1 2 119866

(13)

where Ω = 119903 0 lt 119903 lt 119877 120597Ω = 0 119877 120601119892 in 120597Ω is zeroand 120601119892 is continuous inside of Ω

Assuming Ω is divided into a limited number of subre-gions Space discretization in one-dimensional X geometry isshown in Figure 1

LetΔ119909 be the split size and119873119883 be the split number in theX geometryThemedium in each subdivision area is uniformso the various cross sections and diffusion coefficients areconstant Using a backward difference format we obtain theequations as formulation (14)

119886119892 (119894) 120601119892119898 (119894 minus 1) + 119887119892 (119894) 120601119892119898 (119894) + 119888119892 (119894) 120601119892119898 (119894 + 1)minus 119878119878119892119898 (119894) = 119865119878119892119898 (119894)

119892 = 1 2 119866 119894 = 1 2 119873119883(14)

4 Mathematical Problems in Engineering

Equation (14) is written as an overrelaxation iterative format

120601(119870)119892119896 = 120596119887119892 (119894) [119865119878119892119898minus1 (119894)] + 119878119878(119870minus1)119892119898 (119894)minus 119886119892 (119894) 120601(119870)119892119898 (119894 minus 1) minus 119888119892 (119894) 120601(119870minus1)119892119898 (119894 + 1)+ (1 minus 120596) 120601(119870minus1)119892119898 (119894)

(15)

where superscript119870 is the inner iteration count and 119898 is theouter iteration count of calculation algorithm where

119886119892 (119894) = minus119863119892 (119894 minus 12)Δ119909 (119894 minus 1) (16)

119887 (119894) = 12Σ119905119892 (119894 +

12)Δ119909 (119894)

+ 12Σ119905119892 (119894 minus

12) Δ119909 (119894 minus 1) minus 119886119892 (119894)

minus 119888119892 (119894) (17)

119888119892 (119894) = minus119863119892 (119894 + 12)Δ119909 (119894) (18)

119865119878119892 (119894) = 120594119892 (119894)119866sum1198921015840=1

(120592Σ119891)1198921015840 1206011198921015840119898minus1 (119894) Δ119909 (119894) (19)

119878119878119892119898 (119894) =119866sum1198921015840=1

Σ1198921015840997888rarr119892 (119894) 1206011198921015840 119898 (119894) Δ119909 (20)

Boundary conditions are as follows

119886119892 (119894minus119861) = minus2119863119892 (119894minus119861)Δ119909 (119894minus119861)

120601119892 (119894minus119861 minus 1) = 0(21)

119886119892 (119894+119861) = minus2119863119892 (119894+119861)Δ119909 (119894+119861)

120601119892 (119894+119861 + 1) = 0(22)

Calculation flow of the program algorithm for (13) is shownin Figure 2

32 Richardson Extrapolation Based on the RichardsonExtrapolation method the convergence of the numericalsolution in (1) and (2) was investigated Let 120593ℎ be thenumerical solution of (13) in a suitably finemesh and 120593 be theexact solution of this problemThe Richardson Extrapolationprocedure assumes that the numerical scheme is first-orderaccurate or second-order accurate Considering the Sobolevspace119867120572(Ω) we assume 120593ℎ isin 119867120572(Ω) and we have

1003817100381710038171003817120593 minus 120593ℎ1003817100381710038171003817119867120572(Ω) = 119900 (ℎ2minus120572) (23)

Specifically the numerical solution 120593ℎ is calculated by thesame discretization scheme with three different grid sizesℎ1 = ℎ ℎ2 = ℎ2 and ℎ3 = ℎ4 respectively

If 120572 = 1 the error can be rewritten by

120593ℎ = 120593119890119909119886119888119905 + 1198880ℎ + 119900 (ℎ2) (24)

120593ℎ2 = 120593119890119909119886119888119905 + 1198880ℎ2 + 119900((ℎ2)2) (25)

120593ℎ4 = 120593119890119909119886119888119905 + 1198880ℎ4 + 119900((ℎ4)2) (26)

Equations (24) to (26) yield

1003817100381710038171003817120593ℎ minus 120593ℎ210038171003817100381710038171198671 = 1198880ℎ2 (27)

1003817100381710038171003817120593ℎ2 minus 120593ℎ410038171003817100381710038171198671 = 1198880ℎ4 (28)

Then we get1003817100381710038171003817120593ℎ minus 120593ℎ2100381710038171003817100381711986711003817100381710038171003817120593ℎ2 minus 120593ℎ410038171003817100381710038171198671 cong 2 (29)

If 120572 = 0 we have120593ℎ = 120593119890119909119886119888119905 + 1198881ℎ2 + 119900 (ℎ3) (30)

120593ℎ2 = 120593119890119909119886119888119905 + 1198881 (ℎ2)2 + 119900((ℎ

2)3) (31)

120593ℎ4 = 120593119890119909119886119888119905 + 1198881 (ℎ4)2 + 119900((ℎ

4)3) (32)

and that the mesh is refined or coarsened by a factor of twoSimilarly we have the following

1003817100381710038171003817120593ℎ minus 120593ℎ2100381710038171003817100381711986701003817100381710038171003817120593ℎ2 minus 120593ℎ410038171003817100381710038171198670 cong 4 (33)

Based on the above derivation the numerical solutionconvergence of 1198670 and 1198671 was obtained by RichardsonExtrapolation method During the above analysis processit is very clear that the exact solutions of neutron diffusionequations are eliminated in the derivation so it is not neededto provide the exact solution as the test oracle Moreoverit is very convenient to verify the program by checkingwhether the numerical solution satisfies the convergenceproperties Moreover as the mesh is encrypted the neutronflux and Max-eigenvalues (effective multipliers) convergerespectively which is also a property that can be usedfor verification Based on the convergence properties threeverification criterions are constructed The first and secondone are the numerical solution convergence of 1198670 and 1198671respectively The last one is the Max-eigenvalues convergencewith the mesh refinement

4 Test Experiments

41 SLAB 1D 2G Benchmark Problem The purpose of thetest experiment is to apply the three verification criterions

Mathematical Problems in Engineering 5

Start

m=1

e ＧＮＢ External iteration vector of Arnoldi mminus1

mminus1rArrgmminus1

External iterative fission source FSgmminus1

k = 1

Initializing the inner iteration neutron flux (kminus1)gm

Internal iterative scattering source SS(kminus1)gm

Internal iterative calculation (k)gm

(kminus1)gm = (k)

gm

(k)gm rArr m

m = Mmminus1 = m

Orthogonal basis vector of Arnoldi Vm = [1 2 middot middot middot M]

Up Heisenberg matrix HM

Calculating eigenvalue problems HMy = y

= VM ∙ y

Output

(k)gm minus (kminus1)

gm

(kminus1)gm

le

Figure 2 Calculation flow chart of the algorithm of HARMNOY-2

presented above to verify the program HARMNOY-2 In thisexperiment we select the SLAB 1D 2G benchmark problemas the computational case which is the one-dimensionalslab reactor and double-group diffusion problem [14] Thethickness of the slab is 450cmwith uniformmeshThe reactorcore geometry is shown in Figure 3 and the cross sectionparameters are shown in Table 1

(1) The HARMONY-2 program performs the SLAB 1D2G benchmark calculation The grid is divided into 18equal parts in one-dimensional Cartesian coordinatesystem For comparison with [14] the output resultsof the eigenvalues are set as four modes eigenvalues

(2) Verification testing by three verification criterionsLet neutron flux be 120593ℎ = (120601ℎ1 120601ℎ2) where 120601ℎ1 is the

6 Mathematical Problems in Engineering

Table 1 Parameters of SLAB 1D 2G one-dimensional slab benchmark problem

Energy group 119863119888119898 Σ119905119888119898minus1 VΣ119891119888119898minus1 120594119901120594119889 Σ1997888rarr2119888119898minus11 1264E-00 8154E-03 0000E-00 100E-00 7368E-032 9328E-01 4100E-03 4562E-03 000E-00 7368E-03

Table 2 Comparison of results by the four modes eigenvalues

n 120582-eigenvalue Absolute errorHARMONY-2 Lecture [14]

1 986988E-01 98698E-01 000002 935159E-01 93516E-01 1000E-053 858979E-01 85898E-01 1000E-054 769751E-01 76976E-01 1000E-05

Table 3 The neutron flux values of 119909119898119894119889 in different refined grids

Mesh Element The neutron flux value of119909119898119894119889 for 1th group energy Ratio1 The neutron flux value of119909119898119894119889 for 2th group energy Ratio1

ℎ1 18 6641297145594463E-002 6641294098874310E-002ℎ2 36 6660342485457200E-002 6660331582277153E-002ℎ3 72 6665076278267033E-002 402327280213385 6665082494407264E-002 400712176556248ℎ4 144 6666264561430789E-002 398372454834075 6666253389102599E-002 405750589616633ℎ5 288 6666565458515177E-002 394913485510786 6666562905904372E-002 378297620234630ℎ6 576 6666633289577738E-002 443597775157830 6666641383170335E-002 394403140799427

450cmxN xN

x3x2x1

middot middot middot middot middot middot

Figure 3 Reactor core configuration of SLAB 1D 2G

neutron scalar flux for 1st group energy 120601ℎ2 is theneutron flux for 2nd group energy and ℎ denotesthe size of grid We assume that ℎ1 = 45018 =25119888119898 ℎ2 = ℎ12 ℎ3 = ℎ14 ℎ4 = ℎ18 ℎ5 =ℎ116 and ℎ6 = ℎ132 are presented as differentrefined grids respectively The point 119909119898119894119889 = 225119888119898in X axis is selected as test point HARMONY-2 isexecuted multiple times depending on the differentmeshes The calculated results are postprocessed andthe value of the flux on observed point is taken outand normalized where the 1198771198861199051198941199001 and 1198771198861199051198941199002 denotethe numerical solution convergence of 1198670 and 1198671respectively as follows

1198771198861199051198941199001 =10038171003817100381710038171003817120593ℎ119894 minus 120593ℎ119894+11003817100381710038171003817100381711986701003817100381710038171003817120593ℎ119894+1 minus 120593ℎ119894+210038171003817100381710038171198670

1198771198861199051198941199002 =10038171003817100381710038171003817120593ℎ119894 minus 120593ℎ119894+11003817100381710038171003817100381711986711003817100381710038171003817120593ℎ119894+1 minus 120593ℎ119894+210038171003817100381710038171198671

(34)

42 Test Results and Discussions The calculation resultsof the four modes eigenvalues are shown in Table 2 andcompared with the results of [14] Table 2 indicates that themaximum absolute error between the results of our programand lecture [14] is very small In addition the neutron fluxof 1206011 1206012 in 18 equal elements mesh are shown in Figures5 and 6 respectively which indicate that the distributionof the neutron scalar flux is geometrically symmetric Thethree verification criterions are applied to verification testfor the numerical solution of the multigroup neutron dif-fusion equations Table 3 shows that the convergence of1198670 on the neutron flux value of 119909119898119894119889 point in differentmeshes In Table 3 the convergence order approximates to4 The derivative values of neutron flux on 119909119898119894119889 point indifferent refined grids are listed in Table 4 As the meshis encrypted the results are about 20 order accuracy inTable 4The calculation results ofMax-eigenvalue in differentrefined grids are shown in Table 5 Figure 4 shows that Max-eigenvalue converges with the mesh encryption When themesh is divided into 72 equal parts the 119896119890119891119891 converges toaround 986943E-01 Those results indicate that calculationof HARMONY-2 satisfies the three verification criterionsconstructed by the proposed method in this paper

Mathematical Problems in Engineering 7

Table 4 The derivative values of neutron flux on 119909119898119894119889 in different refined grids

Mesh Element

The derivative ofneutron flux value of119909119898119894119889 for 1th group

energy

Ratio2

The derivative ofneutron flux value of119909119898119894119889 for 2th group

energy

Ratio2

ℎ1 18 080710858E-04 08071363955E-04ℎ2 36 042252071E-04 019102225E+01 04224298523E-04 019106992E+01ℎ3 72 021143797E-04 019983199E+01 02118749068E-04 019937701E+01ℎ4 144 010581871E-04 019981151E+01 01061129701E-04 019966919E+01ℎ5 288 079545852E-05 013302858E+01 07929059044E-05 013382795E+01

Table 5 The calculation results of Max-eigenvalue in different refined grids

Mesh ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6Element 18 36 72 144 288 576Max-eigenvalue 986988E-01 986954E-01 986945E-01 986943E-01 986943E-01 986942E-01

98715times10minus1

98710times10minus1

98705times10minus1

98700times10minus1

98695times10minus1

Max-eigenvalue

Max

-eig

enva

lue

Number of splits6005004003002001000

Figure 4 The convergence of the Max-eigenvalue

Table 2 actually reflects a comparison method which iscommonly used in verification tests Although this method iseasy to perform by comparing with the reference values it isdifficult to get reference values for some newly investigatedprograms Our method not only is more rigorous thantraditional comparison methods but also does not needreference solutions exact solutions or reference values etcThough our method is based on convergence properties ofnumerical solution it is not difficult to get these convergenceproperties from the other second-order elliptic partial differ-ential equations by our method Clearly our method can beapplied to verify the programs of these equations Thereforeour method is very helpful in alleviating oracle problem andimproving the quality of scientific software

5 Conclusion

In this paper we proposed a Richardson Extrapolation-basedverification method which used the Richardson Extrapola-tion on different levels of mesh refinement to estimate theconvergences of the numerical solution and we constructed

the verification criterions based on these properties Com-pared with traditional methods our method did not need theexact solution analysis solutions and reference solutions asexpected values The calculation program of multigroup neu-tron diffusion equation was used as an example of verificationtestThe results show that the numerical convergences accordwith the verification criterion Though the test experimentindicated the validity of our method for alleviating the oracleproblem further research might explore the application ofour method to test other programs for solving second-orderelliptic partial differential equations

Data Availability

(1) The data of HARMONY-2 code used to support thefindings of this study are available from the correspondingauthor upon request (2) The data of Table 1 supporting thecalculation ofHARMONY-2 in this paper are frompreviouslyreported studies and datasets which have been cited in [14]The processed data are available from the correspondingauthor upon request (3) The data of Tables 2 3 4 and 5 are

8 Mathematical Problems in Engineering

Mesh Number in X

Neu

tron

Flux

Neutron Flux

20181614121086420

007

006

005

004

003

002

001

000

minus001

Figure 5 1206011 in grid with 18 equal elements

Mesh Number in X

Neu

tron

Flux

Neutron Flux

20181614121086420

007

006

005

004

003

002

001

000

minus001

Figure 6 1206012 in grid with 18 equal elements

the calculation results of HARMNOY-2 which are used tosupport the findings of this study and are available from thecorresponding author upon request (4) The data of Figures1ndash6 are the calculation results of HARMNOY-2 which areused to support the findings of this study and are availablefrom the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grant No 11805093The authorswould like to thank Dr Xie Jinsen of University of SouthChina for providing advice on program of HARMONY-2

References

[1] W L Oberkampf and T G Trucano ldquoVerification and valida-tion benchmarksrdquoNuclear Engineering and Design vol 238 no3 pp 716ndash743 2008

[2] A A Staf AIAA guide for the verification and validation ofcomputational fluid dynamics simulations American Institute ofAeronautics amp Astronautics USA 1998

[3] E J Weyuker ldquoOn testing non-testable programsrdquo The Com-puter Journal vol 25 no 4 pp 465ndash470 1982

[4] W EHowdenA survey of dynamic analysismethods Universityof Victoria Department of Mathematics 1978

[5] B Butler M Cox A Forbes S Hannaby and P Harris ldquoAmethodology for testing classes of approximation and optimiza-tion softwarerdquo in Quality of Numerical Software IFIP Advancesin Information and Communication Technology pp 138ndash151Springer Boston Mass USA 1997

[6] J Harrison ldquoFormal proofmdashtheory and practicerdquoNotices of theAMS vol 55 no 11 pp 1395ndash1406 2008

[7] P J Roache ldquoCode verification by the method of manufacturedsolutionsrdquo Journal of Fluids Engineering vol 124 no 1 pp 4ndash102002

[8] T Y Chen S C Cheung and S M Yiu ldquoMetamorphic testinga new approach for generating next test casesrdquo Tech RepHKUST-CS98-01 Department of Computer Science HongKong University of Science and Technology Hong Kong 1998

[9] S Segura G Fraser A B Sanchez and A Ruiz-CortesldquoA Survey on Metamorphic Testingrdquo IEEE Transactions onSoftware Engineering vol 42 no 9 pp 805ndash824 2016

[10] M Imgrund ANSYS Verification Manual Swanson AnalysisSystems Inc 1992

[11] P J RoacheVerification and validation in computational scienceand engineering Hermosa Calif USA 1998

[12] N Massarotti M Ciccolella G Cortellessa and A MauroldquoNew benchmark solutions for transient natural convectionin partially porous annulirdquo International Journal of NumericalMethods for Heat amp Fluid Flow vol 26 no 3-4 pp 1187ndash12252016

[13] F Arpino A Carotenuto M Ciccolella G Cortellessa NMassarotti and A Mauro ldquoTransient natural convection inpartially porous vertical annulirdquo International Journal of Heatand Technology vol 34 no 2 pp S512ndashS518 2016

[14] K P Singh S BDegweker R SModak andK Singh ldquoIterativemethod for obtaining the prompt and delayed alpha-modes ofthe diffusion equationrdquo Annals of Nuclear Energy vol 38 no 9pp 1996ndash2004 2011

[15] C J Roy ldquoReview of code and solution verification proce-dures for computational simulationrdquo Journal of ComputationalPhysics vol 205 no 1 pp 131ndash156 2005

[16] P J Roache ldquoVerification of codes and calculationsrdquo AIAAJournal vol 36 no 5 pp 696ndash702 1998

[17] A V Avvakumov P N Vabishchevich A O Vasilev and VF Strizhov ldquoSolution of the 3D neutron diffusion benchmarkby FEMrdquo in Proceedings of the International Conference onLarge-Scale Scientific Computing vol 10665 of Lecture Notes inComputer Science pp 435ndash442 Springer

[18] H Sekimoto Nuclear Reactor Theory Center ofExcellencemdashInnovative Nuclear Energy Systems for SustainableDevelopment of the World (COE-INES) Tokyo Institute ofTechnology Tokyo Japan 2007 Part 2

[19] R Zanette C Z Petersen M Schramm and J R ZabadalldquoA modified power method for the multilayer multigroup two-dimensional neutron diffusion equationrdquo Annals of NuclearEnergy vol 111 pp 136ndash140 2018

[20] C Ceolin M Schramm Vilhena and B E J Bodmann ldquoOnan evaluation of the continuous flux and dominant Eigenvalue

Mathematical Problems in Engineering 9

problem for the steady state multi-group multi-layer neutrondiffusion equationrdquo in Proceedings of the International NuclearAtlantic Conference - INAC 2013 2013

[21] S Han S Dulla and P Ravetto ldquoComputational methodsfor multidimensional neutron diffusion problemsrdquo Science andTechnology of Nuclear Installations vol 2009 Article ID 97360511 pages 2009

[22] A Hebert D Sekki and R Chambon ldquoA User Guide forDONJON Version4rdquo Tech Rep IGE-300 Ecole Polytechniquede Montreal Montreal QC Canada 2013

[23] R B Lehoucq D C Sorensen and C Yang ARPACK UsersrsquoGuide Solution of Large-Scale Eigenvalue Problems with Implic-itly Restarted Arnoldi Methods ISBN 0-89871-407-9 1998httpwwwcaamriceedusoftwareARPACK

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom