Second-Order Domain Decomposition Method for Three-Dimensional Hyperbolic Problems

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1 Iteratioal Mathematical Forum, Vol. 8, 013, o. 7, Secod-Order Domai Decompositio Method for Three-Dimesioal Hyperbolic Problems Youbae Ju Departmet of Applied Mathematics Kumoh Natioal Istitute of Techology Gumi, Gyeogbuk , Korea Abstract I this paper, a o-overlappig secod-order domai decompositio method for solvig three-dimesioal hyperbolic partial differetial equatio is proposed. Ucoditioal stability of the algorithm is aalyzed. Numerical experimets show that the method is stable ad very efficiet. Mathematics Subject Classificatio: 65M06, 65M55 Keywords: three-dimesioal hyperbolic problem, domai decompositio, parallel algorithm, stability, efficiecy 1 Itroductio Hyperbolic partial differetial equatio describes a umber of iterestig physical problems i diverse areas such as: fluid dyamics ad aerodyamics, solid mechaics, astrophysics, the theory of elasticity, optics, electromagetic waves, direct ad iverse scatterig, ad the geeral theory of relativity. I this paper, we cosider the three-dimesioal liear secod-order hyperbolic partial differetial equatio of the form u tt + a(x,y, z, t)u t + b(x,y, z, t)u = c(x, y, z, t)u xx + d(x, y, z, t)u (1) +e(x, y, z, t)u + f(x, y, z, t) defied i {(x, y, z, t) 0 x, y, z 1, 0 t T }, with iitial coditios u(x, y, z, 0) = φ(x, y, z),u t (x, y, z, 0) = ψ(x, y, z) iω, () This paper was supported by Research Fud, Kumoh Natioal Istitute of Techology.

2 31 Youbae Ju ad boudary coditio u(x, y, z, t) =g(x, y, z, t) o the boudary Ω, 0 t T, (3) where Ω is the boudary of Ω = {(x, y, z) 0 x, y, z 1}. There are may implemetatios o hyperbolic problems such as: fiite volume method with grid refiemet techique [, fiite differece scheme for adaptive mesh refiemet [5, two-time level alteratig directio implicit (ADI) fiite volume scheme [9, ad may others. For three-dimesioal hyperbolic problems, there are Galerki alteratig-directio method [6, ucoditioally stable ADI method [7, operator splittig techique [8. However, most of these methods emphasized maily o the stability, ot o the efficiecy. Domai decompositio (DD) is oe of the techiques for solvig partial differetial equatio. The method is very efficiet especially whe a parallel computer is used. I the method, the origial spatial domai is decomposed ito subdomais ad the pde i each subdomai is solved i parallel maer. Ju [3, 4 has proposed ucoditioally stable DD methods for liear secod-order hyperbolic equatios i oe ad two space dimesios. I geeral, implemetatio of DD method to solve high dimesioal hyperbolic problem is somewhat complicated. So far, there is o literature dealig with three-dimesioal hyperbolic problem usig efficiet domai decompositio techique. I this paper, we focus o domai decompositio method to solve three dimesioal hyperbolic problem. The paper is orgaized as follows. I the ext sectio we itroduce ew algorithm ad aalyze its stability. I Sectio 3, umerical results o the performace of the method are reported with respect to accuracy ad efficiecy. The we make cocludig remarks i Sectio 4. Algorithm ad Stability I this sectio, we describe ew domai decompositio algorithm to solve the problem (1) (3). A fiite differece scheme is used to discretize the partial differetial equatio with the cetral differece formulas. We let wi,j,k be the approximated value to the exact value u i,j,k at the grid poit (x i,y j,z k,t ), where Δx = 1,Δy = 1,Δz = 1,Δt = T, for some preset positive itegers I, I J K N J, K, N ad x i = iδx, y j = jδy, z k = kδz, t = Δt. We defie the fiite differece operators at the grid poit (x i,y j,z k,t )aswtt = w i,j,k w i,j,k +w 1 i,j,k, (Δt) wt w = w i,j,k w 1 i,j,k, w Δt xx = w i+1,j,k w i,j,k +w i 1,j,k, w (Δx) = w i,j+1,k w i,j,k +w i,j 1,k, (Δy) = w i,j,k+1 w i,j,k +w i,j,k 1 (Δz). Ad we deote a(x i,y j,z k,t ), b(x i,y j,z k,t ), c(x i,y j,z k,t ), d(x i,y j,z k,t ), e(x i,y j,z k,t ), ad f(x i,y j,z k,t ), by a ijk, b ijk, c ijk, d ijk, e ijk, ad f ijk, respectively.

3 Domai decompositio method for 3D hyperbolic problems 313 It is well-kow [1 that the classical fully explicit three-level fiite differece scheme referred to as the fully explicit scheme (FES) is coditioally stable ad the scheme ca be writte as follows: w tt + a ijkw t + b ijkw ijk = c ijkw xx + d ijkw + e ijkw + f ijk (4) ad the classical fully implicit three-level scheme referred to as the fully implicit scheme (FIS) is ucoditioally stable, which ca be writte as wtt + a t + b ijk = 1 [ c xx + wxx 1 + d 1 [ + w 1 (5) +e 1 [ + w 1 + f ijk. These FES ad FIS are used to be compared with our ew scheme, i this paper. Now we describe ew fiite differece scheme to solve hyperbolic problem (1) (3) usig domai decompositio techique. Suppose, for the simplicity, we decompose the spatial domai ito o-overlappig stripwise subdomais alog the x-directio show i Figure 1. The it is easy to see that the adjacet subdomais share a iterface plae. z y iterface plaes x Figure 1. Stripwise Decompositio I order to solve each divided subdomai problem idepedetly, the values at the iterface poits eed to be estimated i advace. Uder uiformly spaced iterface plaes assumptio, the distace betwee adjacet iterface plaes is 1 if the domai is decomposed ito P subdomais. Let H = 1. The P P usig the cetral differece scheme for u(x, y, z, t), we obtai u xx (x, y, z, t) = u(x+h,y,z,t) u(x,y,z,t)+u(x H,y,z,t) + O(H ) at the iterface plae. We ote that H H is a iteger multiple of Δx. Thus we defie a operator at the iterface plae as ŵxx = w i+ih,j,k w i,j,k +w i IH,j,k, where w H i+ih,j,k ad w i IH,j,k are the approximated values at the adjacet iterface plaes. The we propose, i

4 314 Youbae Ju this paper, that the values at the iterface plaes are obtaied by the followig iterface predictio scheme: wtt + a ijkwt + b ijkwijk = c 1 [ŵ ijk xx +e 1 [ +ŵxx 1 + d 1 [ + w 1 + f ijk. + w 1 After estimatig iterface values, each sub-problem is solved by the fully implicit scheme. The we repeat this process util the last time level. We call the whole procedure the secod-order implicit predictio (SIP) method. Stecils of the method at each time level are give i Figure. The whole SIP scheme is summarized as the followig. <Secod-order Implicit Predictio (SIP) algorithm> Step1: Predict iterface values usig Equatio (6) wtt + a t + b ijk = c ijk 1 [ŵ xx +ŵxx 1+d ijk 1 +e ijk 1 [w + w 1 +f ijk, where ŵ xx ad ŵ 1 xx Step: Solve iterior liear system usig FIS (5) wtt + a t + b ijk = c ijk 1 [w xx + wxx 1+d ijk 1 +e ijk 1 [w + w 1 +f ijk [ w (6) + w 1 are defied earlier [ w Step3: Repeat Step1 ad Step util the last time level + w 1 wi IH,j,k wi,j,k+1 wi,j+1,k H H w w i+ih,j,k i,j,k wi,j 1,k wi,j,k+1 wi,j+1,k wi 1,j,k h h w w i+1,j,k i,j,k wi,j 1,k w i,j,k 1 w i,j,k 1 (a) Iterface (b) Iterior Figure. Stecils of the SIP scheme at each time level Now we provide a theorem for the ucoditioal stability ad trucatio errors of the predictio scheme ad the iterior scheme of the SIP algorithm, respectively. I the theorem, we see the accuracy of the iterface predictio alog the x-directio of the ew method is of secod order H. Theorem.1. The iterface predictio scheme ad the iterior scheme of the SIP method are both ucoditioally stable ad the error terms are w ijk u ijk = O(H +(Δy) +(Δz) +(Δt) ) ad w ijk u ijk = O((Δx) + (Δy) +(Δz) +(Δt) ), respectively.

5 Domai decompositio method for 3D hyperbolic problems 315 Proof. It is well-kow [1 that the fully implicit scheme (FIS) is ucoditioally stable. The SIP iterface scheme is a kid of FIS, i which the step size of the x-directio is H. Thus, the SIP iterface scheme is ucoditioally stable. Suppose the domai is decomposed ito P subdomais. The ŵxx = u xx + O(H ), w = u + O((Δy) ), w = u + O((Δz) ), wtt = u tt + O((Δt) ), ad wt = u t + O((Δt) ). So we ca easily see that the trucatio error is wijk u ijk = O(H +(Δy) +(Δz) +(Δt) ). Note that H is a iteger multiple of Δx. Similarly, the iterior scheme of SIP is FIS itself, which explais the ucoditioal stability. Fially, we clearly see that wijk u ijk = O((Δx) +(Δy) +(Δz) +(Δt) ). Whe we solve the hyperbolic problem usig the fiite differece method, it is importat to approximate the value at the grid poit (x i,y j,z k,t 1 ). We defie w xx = w0 i+1,j,k w0 ijk +w0 i 1,j,k (Δx) The we approximate the value wijk 1 by, w = w0 i,j+1,k w0 ijk +w0 i,j 1,k (Δy) w 1 ijk = w0 ijk +Δt(ψ) ijk, w = w0 i,j,k+1 w0 ijk +w0 i,j,k 1 (Δz). + (Δt) [ c 0 ijk w xx + d 0 ijk w + e 0 ijk w a 0 ijj (ψ) ijk b 0 0 ijk + f ijk 0. 3 Numerical Experimets I this sectio, we preset the results from our umerical experimets o the proposed secod-order implicit predictio (SIP) algorithm. The fully implicit scheme (FIS) is used for our bechmark compariso sice it is a well kow ucoditioally stable scheme. For the umerical experimets, we solve the followig two model problems for differet values of step sizes. The iitial ad boudary coditios ad f(x, y, z, t) are derived from the exact solutio u = e t sih x sih y sih z. All our umerical experimets are carried out o a IBM System x3400 M3 server with Itel Xeo E5630 CPU ruig at.53ghz. 1. Model Problem 1 (MP1): u tt +e x+y+z u t + si (x + y + z)u =(1+x )u xx +(1+y )u +(1+ z )u + f(x, y, z, t). Model Problem 1 (MP): u tt + x u t + 1 x u =(1+x + y + z )(u xx + u + u )+f(x, y, z, t) First, we ivestigate the ucoditioal stability for the SIP scheme. We compare the proposed SIP scheme of 10 subdomais, SIP(10), with the fully explicit scheme (FES) usig Equatio (4) ad the fully implicit scheme (FIS) usig Equatio (5). Table 1 shows the maximum errors of the model problems

6 316 Youbae Ju at the differet values of λ ragig from 3 to 300, where λ =(Δt 16 Δx ) +( Δt Δy ) + ( Δt Δz ). As we ca see i Table 1, FES is coditioally stable ad FIS ad SIP(10) are ucoditioally stable. We should poit out that our SIP(10) scheme is as accurate as FIS or eve more accurate tha FIS. Secod, we aalyze the efficiecy of the proposed SIP scheme. Oe measuremet for the efficiecy of a parallel algorithm is parallel CPU time (PCPU) which is defied by total CPU time (TCPU) divided by the umber of subdomais (P ). Table shows maximum error, TCPU, ad PCPU of the model problems usig the SIP scheme with various P but fixed λ = 3. We see i Table that PCPU decreases sigificatly whe the umber of subdomais P icreases, which meas that SIP(P ) is very efficiet. Table 1. Maximum error of model problems with various λ (Δx =Δy =Δz = MP1 MP Δt λ FES FIS SIP(10) FES FIS SIP(10) 1/ e e e e-3 1/ e e e e-4 1/ e e e e-4 1/ e e e e-4 1/400 3/ e e e e e e-4 Table. Maximum error ad CPU time of SIP with various P (TCPU=Total CPU, PCPU=Parallel CPU, Δx =Δy =Δz =Δt = P 1(=FIS) MP1 Error 0.605e e e e-4 038e e-4 TCPU PCPU MP Error 0.746e e e e e e-4 TCPU PCPU Coclusio I this paper, we proposed a o-overlappig stripwise domai decompositio scheme for solvig three-dimesioal liear secod-order hyperbolic partial differetial equatios. The o-overlappig stripwise decompositio geerates iterface plaes i three-dimesioal problem. The predictio scheme of the proposed method at the iterface plae is a implicit scheme ad of secod order accuracy. Ucoditioal stability of the method was aalyzed ad demostrated usig umerical experimets i this paper. Numerical results show that

7 Domai decompositio method for 3D hyperbolic problems 317 the ew method is as accurate as the classical implicit three-level fiite differece scheme or eve more accurate, but the oe is much more efficiet tha the other. The method performed very well ot oly at costat coefficiet problems, but also at variable coefficiet problems ad sigular problems. Refereces [1 W.F. Ames, Numerical methods for partial differetial equatios, Academic Press, 199. [ W. Heieke, M. Kuik, The aalytical solutio of two iterestig hyperbolic problems as a test case for a fiite volume method with a ew grid refiemet techique, J. Comput. Appl. Math. 14 (008), [3 Y. Ju, A stable oiterative Predictio/Correctio domai decompositio method for hyperbolic problems, Appl. Math. Comput. 16 (010), [4 Y. Ju, A ucoditioally stable implicit high-order domai decompositio method for two-dimesioal secod-order hyperbolic problem, Appl. Math. Sci. 6 (01), [5 R.M.J. Kramer, C. Patao, D.I. Pulli, A class of eergy stable, highorder fiitedifferece iterface schemes suitable for adaptive mesh refiemet of hyperbolic problems, J. Comput. Phys. 6 (007), [6 X. Lai, Y. Yua, Galerki alteratig-directio method for a kid of three-dimesioal oliear hyperbolic problems, Comput. Math. Appl. 57 (009), [7 R.K. Mohaty, M.K. Jai, U. Arora, A ucoditioally stable ADI method for the liear hyperbolic equatio i three space dimesios, It. J. Comput. Math. 79 (00), [8 R.K. Mohaty, A operator splittig techique for a ucoditioally stable differece method for a liear three space dimesioal hyperbolic equatio with variable coefficiets, Appl. Math. Comput. 16 (005), [9 M. Yag, Two-time level ADI fiite volume method for a class of secodorder hyperbolic problems, Appl. Math. Comput. 15 (010), Received: October, 01