AN EFFICIENT IMPLEMENTATION OF IMPLICIT OPERATOR FOR BLOCK LU-SGS METHOD

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1 Computatioal Fluid Dyamics JOURNAL 13(): July 5 (pp ) AN EFFICIENT IMPLEMENTATION OF IMPLICIT OPERATOR FOR BLOCK LU-SGS METHOD Joo Sug KIM Oh Joo KWON Abstract A efficiet implemetatio techique of a block matrix implicit operator based o va Leer s FVS scheme is proposed for the LU-SGS method o ustructured meshes. The off-diagoal terms of the liear system of equatios were approximated by the icremet of the umerical flux usig the poit-by-poit splittig feature of FVS schemes. The preset improved LU-SGS method provides better performace i terms of CPU time ad requires substatially less storage tha the correspodig baselie method. The covergece rate of the preset improved method is faster tha the improved method based o the Jameso ad Turkel implicit operator. Key Words: Implicit operator, LU-SGS method, Flux-vector splittig, Euler equatios 1 INTRODUCTION For the robustess ad efficiecy of umerical calculatios, implicit schemes usually adopt a implicit operator (left-had side) that has a umerical dissipatio level higher tha that of the explicit operator (right-had side) [1]. Whe Roe s flux-differece splittig (FDS) scheme [] is used as the explicit operator, the flux-vector splittig (FVS) scheme of Steger ad Warmig [3] or va Leer [4] ca be a viable choice as the implicit operator. A cosistet scheme, such as usig Roe s FDS scheme for both implicit ad explicit operators, may result i severe stability ad covergece degradatio. The lower-upper symmetric Gauss-Seidel (LU-SGS) method was origially developed by Jameso ad Yoo [5] o structured meshes. This method was successfully geeralized to ustructured meshes i a way that a large portio of memory requiremet ca be reduced while retaiig comparable computatioal time [6]. I this study, a implicit operator proposed by Jameso ad Turkel [7] was approximated to a form ivolvig the icremet of the flux vector ad icorpo- Received o Jue, 5. Doctoral cadidate, Departmet of Aerospace Egieerig, Korea Advaced Istitute of Sciece ad Techology, Daejo, Korea Correspodig Author, Professor, Departmet of Aerospace Egieerig, Korea Advaced Istitute of Sciece ad Techology, Daejo, Korea, ( ) : ojkwo@kaist.ac.kr rated ito the origial LU-SGS method. Eve though sigificat memory reductio was achieved through this implemetatio, the problem of slow covergece of the Jameso ad Turkel implicit operator remaied [8, 9]. I the preset study, a techique for the efficiet implemetatio of a block matrix implicit operator based o va Leer s FVS scheme is proposed for the LU- SGS method o ustructured meshes. To reduce the memory requiremet ad the computatioal time simultaeously, the off-diagoal terms of the discretized liear system of equatios were approximated by the icremet of the umerical flux usig the poit-bypoit splittig feature of FVS schemes. As a explicit operator, Roe s FDS scheme was used to maitai the solutio accuracy. Validatios were made by applyig the preset method to a freestream flow ad the flow over a RAE 8 airfoil. IMPLICIT FINITE-VOLUME METHOD The two-dimesioal compressible Euler equatios with a ideal gas assumptio may be writte i a itegral formulatio for a cotrol volume Ω with the boudary Ω: QdV+ f(q) dl = (1) t Ω Ω where Q represets the vector of depedet variables ad f(q) is the iviscid flux vector. The goverig equatios were dicretized usig a ode-based fiitevolume method i which the variables were stored at

2 A Efficiet Implemetatio of Implicit Operator for Block LU-SGS Method 155 the odes of the mesh. The computatioal domai was divided ito a fiite umber of triagles, ad the cotrol volume was costructed from the media dual mesh surroudig each ode [1]. The applicatio of the Euler implicit scheme ad the local time liearizatio at time level yields a liear system of equatios: LΔQ = R () where ΔQ = Q +1 Q, ad R ad L represet the explicit ad implicit operators, respectively, ad are writte for ode i as R i = F ij (Q l,q r )l ij (3) j f(i) L i = V i I+ R i Δt i Q (4) Here, f(i) is the set of face eighbor odes, l ij is the legth of the jth boudary of the cotrol volume, V i is the area of the cotrol volume, ad Δt i is the time step. Also, F ij (Q l,q r ) is the umerical iviscid flux where the subscripts l ad r refer to the left ad right sides of the boudary, respectively. The Jacobia R i / Q of the implicit operator was approximated by the first-order accuracy rather tha by exactly liearizig the higher-order accurate explicit operator. This defect correct algorithm [11] is commo to may implicit codes because it reduces the badwidth ad ehaces the diagoal domiace property of the associated liear system of equatios. I this approximate approach, cosistet splittig schemes for the implicit ad explicit sides are ot required, sice the implicit side does ot affect the spatial accuracy of the steady-state solutio. Furthermore, cosistet schemes may ot ecessarily exhibit a better performace tha other possible icosistet combiatios of the implicit ad explicit sides. For the first-order accurate differecig, the data at each cotrol volume face is set equal to the oe at the correspodig ode. For the high-order differecig, the primitive variables at each cotrol volume face are liearly extrapolated from surroudig odes usig a Taylor series expasio. The solutio gradiet required for the expasio is calculated usig a uweighted least-square procedure [1]. This high-order differecig is equivalet to a secod-order, upwid biased MUSCL differecig o orthogoal structured grids [1]. Vekatakrisha s limiter [13] is used to cotrol udesirable oscillatios aroud discotiuities. 3 BASELINE LU-SGS METHOD The i-th row of the resultat liear system of equatios ca be writte as a liear combiatio of the diagoal ad off-diagoal elemets: [D ii ]ΔQ i + [O ij ]ΔQ j j L(i) + j U(i) [O ij ]ΔQ j = R i (5) where L(i) ad U(i) represet the lower ad upper eighbor odes of ode i, ad [D ii ] ad [O ij ] are give by [D ii ]= V i Δt i + j f(i) F ij Q i (6) [O ij ]= F ij (7) Q j The the liear system ca be solved usig multiple lower ad upper sweeps of symmetric Gauss-Seidel (LU-SGS) method as show i the followig. Lower sweep: [D ii ]ΔQ i + [O ij ]ΔQ j j L(i) + j U(i) Upper sweep: [D ii ]ΔQ k i + [O ij ]ΔQ j j L(i) + j U(i) [O ij ]ΔQ k 1 j = R i (8) [O ij ]ΔQ k j = R i (9) where the superscript k is the subiteratio idex ad ΔQ is take to be zero. Also, the superscript is used to represet the most recet value obtaied from the lower sweep. I the LU-SGS method of Jameso ad Yoo [5] the flux fuctio i the implicit side is chose as F ij = 1 [f ij(q i )+f ij (Q j ) ρ ij (Q j Q i )] (1) where ρ ij is the spectral radius of the iviscid flux Jacobia at the jth cotrol volume boudary. The the Jameso ad Turkel implicit operator [7] is obtaied by liearizig the flux fuctio (1): F ij Q i = 1 [ fij (Q i ) Q i + ρ ij I ] (11)

3 156 Joo Sug Kim ad Oh Joo Kwo F ij = 1 [ ] fij (Q j ) ρ ij I Q j Q j (1) Whe icorporated with the LU-SGS method, the Jameso ad Turkel implicit operator is early ucoditioally stable [14]. However, it was also show that the implicit operator based o the liearizatio of the FVS scheme of Steger ad Warmig or va Leer leads to a superior covergece rate to the Jameso ad Turkel implicit operator [8]. The umerical flux of FVS schemes is usually writte as a sum of the forward flux F + ad the backward flux F [4]: F ij = F + ij (Q i)+f ij (Q j) (13) The liearizatio procedure for FVS schemes ca be easily accomplished sice the splittigs are carried out locally idepedet of eighborig poits. I the preset study, oly va Leer s FVS scheme is cosidered for the implicit operator because of its superior covergece rate over Steger ad Warmig s FVS scheme [1]. I the symmetric Gauss-Seidel iteratios, iformatio about the liear system is usually stored i the computer memory usig the compressed sparse-row format. The diagoal ad off-diagoal elemets require a storage of ode eq eq ad edge eq eq, respectively, i the preset odebased fiite-volume method. Here ode, edge ad eq are the umber of odes, the umber of edges ad the umber of ukow variables, respectively. Sice edge is approximately equal to 3 ode for two-dimesioal triagular meshes ad 7 ode for three-dimesioal tetrahedral meshes, the off-diagoal terms occupy approximately 86% of the total memory requiremet i two dimesios ad 93% i three dimesios [15]. 4 IMPROVED LU-SGS METHOD For the Jameso ad Turkel implicit operator, the matrix-vector product terms i eq. (5) ca be approximated as [6, 9] [D ii ]ΔQ i V i ΔQ i Δt i + ( ( Fij Q i +ΔQ i, ) Q j Fij (Q i, Q j )) j f(i) = V i + 1 Δt i [O ij ]ΔQ j F ij j f(i) ρ ij ΔQ i (14) ( Q i, Q ) j +ΔQ j Fij (Q i,q j ) = 1 [ ( fij Q j +ΔQ ) j fij (Q j ) ] ρ ij ΔQ j (15) Now, the diagoal block matrix is simplified to the idetity matrix with a scalar factor, ad the product of the off-diagoal block matrix ad the vector of flow variable icremet is also reduced to the icremet of the iviscid flux vector. By adoptig this approximatios, the block matrix of the liear system does ot have to be stored, ad a storage of ode is oly required for the scalar factor of the diagoal term. This memory reductio is approximately equivalet to 99.1% ad 99.7% cut from the memory requiremet of the baselie method for two ad threedimesioal ( liear ) systems, respectively, eve though f ij Q j +ΔQ j ad ρ ij ΔQ j i the off-diagoal term eed to be repeatedly evaluated durig subiteratio. For va Leer s FVS scheme, the product of the offdiagoal matrix ad the vector of flow variable icremet i eq. (5) ca be writte as [O ij ]ΔQ j F ij ( Q j +ΔQ j ) F ij (Q j ) (16) Here the forward flux F + is elimiated due to the poit-by-poit splittig property of FVS schemes. I this implemetatio, sice the diagoal term is ot modified, oly a storage of ode eq eq is required for storig the diagoal block matrix, which correspods to approximately 14% ad 7% of the total required memory for storig the baselie liear system of equatios i two ad three dimesios, respectively. To achieve this memory requiremet reductio, the term Fij ( ) Q j +ΔQ j correspodig to the offdiagoal elemet should be repeatedly evaluated durig subiteratio. However, the computatioal time of this overhead per a set of lower ad upper sweeps is approximately equivalet to a half of that of the firstorder accurate flux computatio, which is less tha few percets of the total CPU time for the preset implicit ode-based ustructured mesh flow solver. The improved method does ot require ay special storage scheme such as the compressed sparse-row format, sice oly the diagoal elemet is stored i the computer memory for each ode. Because of this simple data structure, the possibility of cache memory misses ad idirect addressig is sigificatly reduced compared to that of the baselie method. 5 NUMERICAL RESULTS I order to compare the performace of the baselie ad improved LU-SGS methods, represetative umerical cases were tested o ustructured triagular meshes. All calculatios were carried out o a Liux-

4 A Efficiet Implemetatio of Implicit Operator for Block LU-SGS Method 157 based PC with a Petium IV.4GHz processor. correspodig baselie method. 1 Baselie with va Leer s FVS Baselie with Jameso ad Turkel s operator Improved with va Leer s FVS 8 Improved with Jameso ad Turkel s operator Fig. 1: Ustructured triagular mesh for freestream flow. 5.1 Freestream Flow I order to assess the effect of the umber of subiteratios ad the umber of grid poits o the computatioal performace, a simple freestream flow was tested. The freestream Mach umber was set to.5 ad the flow agle was zero. The computatioal domai was uit legths log ad 1 uit legths high, ad o disturbace was give iside the flow field. The triagular mesh for the preset calculatio cosisted of 6, 83 odes ad 13, 63 cells (Fig. 1). CPU time/time step Baselie with va Leer s FVS Baselie with Jameso ad Turkel s operator Improved with va Leer s FVS Improved with Jameso ad Turkel s operator No. of subiteratios Fig. : Compariso of CPU time per time step i terms of umber of subiteratios for freestream flow. The effect of evaluatig the off-diagoal terms repeatedly durig subiteratio o CPU time is preseted as a fuctio of umber of subiteratios i Fig.. It shows that the preset implemetatio based o va Leer s FVS scheme requires approximately 13% less CPU time tha the correspodig baselie method for all subiteratios. Meawhile, the improved method based o the Jameso ad Turkel implicit operator requires approximately 8% less CPU time tha the CPU time/time step No. of odes( 1 4 ) Fig. 3: Compariso of CPU time per time step i terms of umber of odes for freestream flow. Next, the computatioal performace is evaluated for varyig umber of grid poits. I Fig. 3, the CPU time per time step is preseted for meshes with the umber of odes from 6, 83 to 176, 541. It shows that the preset implemetatio based o va Leer s FVS scheme cosistetly requires approximately 14% less CPU time per time step tha the correspodig baselie method. The improved method based o the Jameso ad Turkel implicit operator showed approximately 5% less CPU time tha the correspodig baselie method. From the above results, it ca be cocluded that, i spite of the slight icrease i the umber of floatig poit operatios, the improved method is always faster ad requires substatially less storage tha the baselie method by efficietly utilizig the cache memory ad reducig the idirect addressig. 5. RAE 8 Airfoil As a secod test case, the flow over a RAE 8 airfoil at a trasoic Mach umber of.75 ad a agle of attack of 3 o was calculated. A ustructured mesh cotaiig 5, 159 odes ad 1, 5 triagular cells was used for the calculatio. A ear field view of the mesh is show i Fig. 4. I Fig. 5, the computed pressure coefficiet is compared with the experimetal data [16]. It shows that the computatioal predictio agrees very well with the experimetal data. I Figs. 6 ad 7, the covergece histories of the baselie ad improved methods are preseted i terms of umber of time steps ad CPU time. The CFL umber ad the umber of subiteratios were adjusted

5 158 Joo Sug Kim ad Oh Joo Kwo Baselie with va Leer s FVS Baselie with Jameso ad Turkel s operator Improved with va Leer s FVS Improved with Jameso ad Turkel s operator Log(Res) Fig. 4: Ustructured triagular mesh for RAE 8 airfoil flow No. of time steps Fig. 6: Compariso of covergece rates i terms of umber of time steps for RAE 8 airfoil flow. -.5 C p Computatio Experimet x/c Fig. 5: Compariso betwee computed ad experimetal pressure coefficiets for RAE 8 airfoil flow. i a way that a best covergece rate ca be obtaied for each method i terms of CPU time. For the va Leer s FVS scheme, the CFL umber was icreased from 1 1 to 1 3 by followig a expoetioal fuctio of the iverse of the residual orm ad seve subiteratios were used. For the compared Jameso ad Turkel implicit operator, the CFL umber was icreased from 1 1 to 1 7 ad the same umber of subiteratios was used. It was observed that each improved method shows a very similar covergece rate to its correspodig baselie method i terms of time step. However, i terms of CPU time, the improved method based o va Leer s FVS scheme shows a covergece rate approximately % faster tha the correspodig baselie method. Meawhile, the im- Log(Res) Baselie with va Leer s FVS Baselie with Jameso ad Turkel s operator Improved with va Leer s FVS Improved with Jameso ad Turkel s operator CPU time Fig. 7: Compariso of covergece rates i terms of CPU time for RAE 8 airfoil flow. proved method based o the Jameso ad Turkel implicit operator coverged approximately 3% faster tha the correspodig baselie method. Overall, the improved method based o va Leer s FVS scheme coverged approximately two times faster tha that of the improved method based o the Jameso ad Turkel implicit operator.

6 A Efficiet Implemetatio of Implicit Operator for Block LU-SGS Method CONCLUSIONS A efficiet implemetatio techique for the LU- SGS method has bee developed for solvig the Euler equatios o ustructured meshes usig va Leer s FVS scheme as a implicit operator. The off-diagoal terms of the liear system of equatios were approximated as the icremet of the flux vector by usig the poit-by-poit splittig property of FVS schemes. It was foud that the preset improved LU-SGS method provides better performace i terms of CPU time ad requires substatially less storage tha the correspodig baselie method. Furthermore, the covergece rate of the preset improved method is much faster tha the improved method based o the Jameso ad Turkel implicit operator. REFERENCES [1] M. S. Liou ad B. va Leer, Choice of implicit ad explicit operators for the upwid differecig method, AIAA Paper 88 64, [] P. L. Roe, Approximate Riema solvers, parameter vectors, ad differece schemes, Joural of Computatioal Physics, Vol. 43, 1981, pp [3] J. L. Steger ad R. F. Warmig, Flux vector splittig of the iviscid gasdyamic equatios with applicatio to fiite-differece methods, Joural of Computatioal Physics, Vol. 4, 1981, pp [4] B. va Leer, Flux-vector splittig for the Euler equatios, Lecture Notes i Physics, Vol. 17, 198, pp [5] A. Jameso ad S. Yoo, Lower-upper implicit schemes with multiple grids for the Euler equatios, AIAA Joural, Vol. 5, No. 7, 1987, pp [6] D. Sharov ad K. Nakahashi, Reorderig of 3- d hybrid ustructured grids for vectorized LU- SGS Navier-Stokes computatios, AIAA Paper 97 1, [7] A. Jameso ad E. Turkel, Implicit schemes ad LU decompositios, Mathematics of Computatio, Vol. 37, No. 156, 1981, pp [8] M. J. Wright, G. V. Cadler ad M. Prampolii, Data-parallel lower-upper relaxatio method for the Navier-Stokes equatios, AIAA Joural, Vol. 34, No. 7, 1996, pp [9] R. F. Che ad Z. J. Wag, Fast, block lowerupper symmetric Gauss-Seidel scheme for arbitrary grids, AIAA Joural, Vol. 38, No. 1,, pp [1] W. K. Aderso ad D. L. Bohaus, A implicit upwid algorithm for computig turbulet flows o ustructured grids, Computers ad Fluids, Vol. 3, No. 1, 1994, pp [11] B. Kore, Defect correctio ad multigrid for a efficiet ad accurate computatio of airfoil flows, Joural of Computatioal Physics, Vol. 77, 1988, pp [1] W. K. Aderso, J. L. Thomas ad B. va Leer, Compariso of fiite volume flux vector splittigs for the Euler equatios, AIAA Joural, Vol. 4, No. 9, 1986, pp [13] V. Vekatakrisha, Covergece to steady state solutios of the Euler equatios o ustructured grids with limiters, Joural of Computatioal Physics, Vol. 118, 1995, pp [14] S. Yoo ad A. Jameso, Lower-upper symmetric-gauss-seidel method for the Euler ad Navier-Stokes equatios, AIAA Joural, Vol. 6, No. 9, 1988, pp [15] H. Luo, J. D. Baum ad R. Löher, A fast, matrix-free implicit method for compressible flows o ustructured grids, Joural of Computatioal Physics, Vol. 146, 1998, pp [16] AGARD Test cases for iviscid flow field method, AGARD advisory report, 11, 1986.