P-MULTIGRID PRECONDITIONERS APPLIED TO HIGH-ORDER DG AND HDG DISCRETIZATIONS

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1 6th European Conference on Computationa Mechanics (ECCM 6) 7th European Conference on Computationa Fuid Dynamics (ECFD 7) June 2018, Gasgow, UK P-MULTIGRID PRECONDITIONERS APPLIED TO HIGH-ORDER DG AND HDG DISCRETIZATIONS M. FRANCIOLINI 1,2, K. J. FIDKOWSKI 1 and A. CRIVELLINI 2 1 Department of Aerospace Engineering, University of Michigan 1320 Bea Ave, 48109, Ann Arbor (MI), USA {mfrancio,kfid}@umich.edu 2 Department of Industria Engineering and Mathematica Science, Poytechnic University of Marche Via Brecce Bianche 12, 60131, Ancona, Itay {m.francioini@pm.,a.criveini@}univpm.it Key words: high-order, DG, HDG, p-mutigrid, preconditioners, parae efficiency Abstract. In this work the use of a p-mutigrid preconditioned fexibe GMRES sover to dea with the soution of stiff inear systems arising from high order time discretization is expored in the context of two high-order spatia discretizations. The first one is a standard moda discontinuous Gaerkin method, whie the second one is an hybridizabe discontinuous Gaerkin method, which for high order has fewer gobay-couped degrees of freedom compared to DG. The efficiency of the proposed soution strategy is assessed on ow-mach, two-dimensiona, compressibe fow probems. The numerica resuts highight that a considerabe reduction in the number of GMRES iterations can be achieved for both space discretizations, but that ony with DG is this gain refected in the CPU time. Moreover, a comparison of the performance shed ight on the convenience of using the former or the atter space discretization. 1 INTRODUCTION In recent years, high-order discontinuous Gaerkin (DG) methods have become increasingy popuar in the fied of Computationa Fuid Dynamics. This fact is certainy ascribed to their convenient dispersion and diffusion properties, the ease of paraeization thanks to their compact stenci, and their accuracy in arbitrary compex geometries. However, the impementation of an efficient soution strategy is sti a subject of active research, especiay for unsteady fow probems [1] invoving the soution of the Navier Stokes (NS) equations. Severa previous studies demonstrated that impicit schemes in the context of highorder space discretizations are one of the most viabe ways to overcome the strict stabiity imits of expicit time integration schemes [2]. Such strategies require the soution of a arge system of inear/non-inear equations, which is typicay performed with iterative sovers such as the generaised minimia residua method (GMRES). The choice of the

2 preconditioner is a key aspect of the strategy and it has been expored extensivey in the iterature, see for exampe [3, 4]. Among those, the use of mutieve agorithms to precondition a fexibe GMRES sover has been demonstrated to be an appeaing choice for both compressibe [3] and incompressibe fow probems [5]. More recenty, hybridizabe discontinuous Gaerkin (HDG) methods have been considered as an aternative to the standard discontinous Gaerkin discretization [6, 7]. HDG methods, which introduce an additiona trace variabe defined on the mesh eement faces, can reduce the gobay couped degrees of freedom if compared to DG when a high order of poynomia approximation is empoyed. In fact, expoiting the bock-structured nature of the matrix, the system size can be reduced using static condensation. Moreover, HDG methods exhibit superconvergence properties of the gradient variabe in diffusiondominated regimes. On the other hand, they increase the amount of operations oca to each eement and the workoad between and after the inear system soution. Whie severa works aimed to compare the accuracy of HDG both versus CG and DG, a comparison of the iterative sovers is missing in this context. Additionay, the use of mutieve soution strategies in HDG contexts has been introduced ony partiay by [8], in the context of an h-mutigrid strategy with trace variabe projections on an underying continuous finite eement space. The present work focuses on the inear soution process. In particuar, the use of a p-mutigrid preconditioned fexibe GMRES agorithm to dea with the soution of stiff inear systems arising from a high-order time discretization is expored in the context of DG and HDG high-order spatia discretizations. The scaabiity of the agorithm is aso considered and compared to standard singe-grid preconditioners ike ILU(0). The efficiency of the different soution strategies is assessed on two-dimensiona aminar compressibe fow probems. The resuts of such cases suggest that (i) simiar error eves can be obtained by the two sovers, (ii) the use of a mutieve strategy reduces consideraby the number of inear iterations, and (iii) ony for the DG discretizations this advantage is refected in the CPU time. 2 DISCRETIZATION The set of compressibe Navier Stokes (NS) equations can be written in compact form as u + F(u, u) = 0, (1) t where u R M is the state vector and F R M N is the sum of the inviscid and viscous fuxes, with M being the number of conservative variabes and N the number of space dimensions. For HDG, (1) is written as a system of first-order partia differentia equations, by introducing q R M N such that q u = 0, u t + F(u, q) = 0. (2) The space and time discretization is outined in the foowing sections. 2

3 2.1 Spatia This work considers two impementations of the discontinuous Gaerkin finite eement method. The first one is a moda-based DG sover operating on a trianguation T h of the domain Ω. The state vector is approximated by a poynomia expansion with no continuity constraints imposed between adjacent eements: u h [V h ] M where V h = {u L 2 (Ω) : u K P k, K T h } and k is the order of poynomia approximation. The weak-form of (1) foows from mutipying the PDE by test functions in the same approximation space, integrating by parts, and couping eements via consistent and stabe numerica fuxes. By foowing this procedure a system of ordinary differentia equations for the degrees of freedom (DoFs) of the probem can be written in the form M du dt + R = 0, (3) where M is the mass matrix, U the vector of DoFs and R the residuas vector. The second spatia discretization is the HDG method [7]. The HDG discretization approximates the variabes u h, q h with u h [V h ] M and q h [V] M N. Moreover, an additiona variabe λ h [M h ] M is defined in the space M h = {λ L 2 (F h ) : λ σf P k, σ f F h }, where F h is the set of interior faces σ f, and P k is the space of poynomias of order k on face σ f. The weak form is obtained in this case by weighting the equations in (2) with appropriate test functions, integrating by parts, and using the interface variabe λ h for the face state. A consistent and stabe fux function is introduced at the mesh eement interfaces, where continuity of the fux is ensured by additiona equations required to cose the system. See [7] for further detais. Defining R Q, R U and R Λ the residuas vectors arising from the gradient equation, NS equations and fux-consistency equations, the ODE system of equations can be written as R Q = 0, M U du dt + RU = 0, (4) R Λ = 0. where M U is the eementa mass matrix. Defining the soution vector of the discrete unkown of the approximation as W = [Q; U; Λ], and the vector of aggomerated residuas R = [R Q ; R U ; R Λ ], the compact form of (4) becomes M dw dt + R = 0, (5) where the matrix M and the vector W foow directy from equation (4). It is worth noticing that, for a DG discretization, W = U and M = M U. 3

4 2.2 Tempora The tempora discretization is an expicit-singe-diagona-impicit Runge Kutta (ES- DIRK) scheme. The genera formuation of the scheme for equation (5) is MW i = MW n t W n+1 = W n + i a ij R(W j ), j=1 s β i tw i, for i = 1,..., s where s is the number of stages, a ij and b i are the coefficients of the scheme, and n is the time index. Within each stage, the soution of a non-inear system is required; to this end, the Newton-Kryov method is used and the k th Newton Kryov iteration assumes the form ( M a ii t + R ) (Wk+1 i W i W k) = M i 1 a ii t (Wi k W n a ij ) R(W j ) R(W a k), i (7) ii with i = 1,..., s. In this work the order-three ESDIRK3 scheme [9] wi be empoyed. 3 LINEAR SOLVER For a DG discretization, system (7) is soved iterativey without any additiona operations, and assumes the form Ax + b = 0, (8) with A = (M/(a ii t)+ R/ W) the iteration matrix, x = W the vector of the degrees of freedom update and b the right-hand side. On the other hand, the HDG discretization expoits the introduction of face unknowns in order to reduce the size of the matrix to be aocated, see [7] for further detais. In fact, system (7) can be conveninety arranged using the definition of gradients residuas, state residuas and trace variabes residuas as A QQ A QU B QΛ A UQ A UU B UΛ C ΛQ C ΛU D Q U Λ i=1 + j=1 b Q b U b Λ (6) = 0, (9) where the bock-structure of the iteration matrix and the right-hand side can be easiy derived from (4). The soution of the system is obtained by staticay condensing out the eement-interior variabes, resuting in the foowing probem ( D [ C ΛQ C ] [ ] ΛU A QQ A QU 1 [ ] ) B QΛ Λ+ ( A UQ A UU b Λ [ C ΛQ C ] [ ΛU A QQ A QU A UQ A UU 4 B UΛ ] 1 [ b Q b U ] ) = 0, (10)

5 which assumes the same form as system (8). It is worth noticing that in HDG the memory aocation and the time spent on goba sove are ower than that of a DG sover due to the smaer amount of gobay couped degrees of freedom. On the other hand, the number of eementa operations is higher. In fact, the inversion of the A ij bock-structured matrix of equation (10), athough being oca to each eement is sti not a trivia cost. Note that the eement-interior states have to be recovered after the inear sove to evauate the residuas vector in the successive iteration using a back-sove strategy. See [7] for further detais. The inear system of ODEs can be soved numericay using iterative sovers. To this end, we empoy the generaized minima residua method (GMRES). Three preconditioner matrices are considered in the remaining of the paper to speed-up the iterative process. The first one is eement-wise bock-jacobi (BJ), which extracts the bock-diagona part of the iteration matrix and takes the LU factorization in a oca-to-each eement fashion. The second one [10] is ine-jacobi (LJ), which creates within the mesh ines of eements of maximum couping and soves impicity the degrees of freedom for each ine. The ast one is the incompete ower-upper factorization with zero-fi, ILU 0 (A), and minimum discarded fi reordering [11]. When appied in parae, the ILU 0 (A) is performed on each square, partition-wise bock of the iteration matrix. An important characteristics of GMRES is that any iterative inear sover, such as mutigrid, can be used as a preconditioner. In such case, the fexibe impementation of GMRES wi be empoyed. 4 P-MULTIGRID PRECONDITIONING In this work, the use of a p-mutigrid strategy is expored in the context of the space discretizations presented. Coarser inear systems A i x i = b i are buit using ower-order poynomia spaces. Subspace inheritance [5] is used to generate coarse grid operators, i.e. the matrices A i, both for the DG and HDG discretizations. This choice invoves projection of the iteration matrix, which is computed ony once on the finest eve. Compared to subspace non-inheritance, which requires the recomputation of the Jacobian in proper coarser-space discretizations of the probem, inheritate is cheaper in processing and memory. Athough previous work has shown ower convergence rates when using such cheaper operators [12], especiay in the context of eiptic probems and incompressibe fows, we found these operators efficient enough for our target probems invoving the compressibe NS equations. The mutigrid strategy requires the definition of both restriction and proongation operators to project the vectors of degrees of freedom between the poynomia spaces. To do so, a distinction between the DG and HDG space discretization has to be performed. In fact, whie in the former case the inear sover works with eement-wise unknowns, in the atter the eement-interior coefficients are staticay condensed out, and the system is soved for the face unknowns ony. As regards DG, et us define a sequence of approximation spaces V V h on the same trianguation T h, being a mutigrid eve, with V h = V 1 V 2... V NLV L and N LV L the tota number of eves. Note that V NLV L denotes the coarsest space. In this context, the proongation operator can be defined as 5

6 I+1 : V +1 V such that ( I +1 u +1 u +1 ) = 0, u+1 V +1. (11) K T h K Simiary, the restriction operator can be defined as the L 2 projection I +1 : V V +1 such that ( ) I +1 u u v+1 = 0, (u, v +1 ) V V +1. (12) K T h K Such definitions can be easiy extended to buid discrete matrix operators I j i that project the vectors and matrices in the coarser eves. Simiar considerations can be done for the face unknowns of the HDG discretization. In this case the sequence of approximation spaces M M h is propery defined on the interior mesh eement faces F h, being a mutigrid eve. To this end, et us define M h = M 1 M 2... M NLV L. The proongation operator is now I+1 : M +1 M such that ( ) I +1 λ +1 λ +1 = 0, λ+1 M +1. (13) F F h F On the other hand, the restriction is defined as I +1 : M M +1 such that ( ) I +1 λ λ µ+1 = 0, (λ, µ +1 ) M M +1. (14) F F h F These definitions can aso be extended to buid matrix operators I j i that transform the vectors, residuas, and bock matrices shown in equation (10). Note that the projection of the fine-space condensed matrix and the right hand side differs from performing a static condensation of the projected matrices and vectors in coarse space. Despite this discrepancy, the resuts show that a consideraby arge reduction of the number of iterations is achieved. We empoy a fu mutigrid (FMG) V-Cyce sover, outined in Agorithm 1. The FMG cyce constructs a good initia guess for a V-Cyce iteration which starts on the fine grid. To do so, the soution is initiay soved on the coarsest eve (N LV L ), and then proongated to the next refined one (N LV L 1). At this point, a standard V-Cyce is caed, such that an improved approximation of the soution can be used for the V-Cyce at eve N LV L 2. This procedure is repeated unti the V-Cyce on the finest eve is competed. The singe V- Cyce is outined in Agorithm 2. Starting from a eve, the soution is initiay smoothed using an iterative sover (SMOOTH). The residua of the soution r is then computed and projected in the coarser eve + 1, where another V-Cyce is recursivey caed to obtain a coarse-grid correction e +1. The quantity is then proongated on to eve and used to correct the soution to be smoothed again. When the coarsest eve is reached, the probem is soved with a higher number of iterations to decrease as much as possibe the soution error. 6

7 Agorithm 1 F MG 1: for = N LV L, 1, 1 do 2: if = N LV L then 3: b = I 1b 1 4: SOLVE A x F MG = b 5: ese 6: b = I 1b 1 7: x 0 = I MG +1xF+1 8: x F MG = MG V (, b, x 0 ) 9: end if 10: end for 11: return x F MG Agorithm 2 MG V (, b, x ) 1: if = N LV L then 2: SOLVE A x = b 3: ese 4: x =SMOOTH(x, A, b ) 5: r = b A x 6: r +1 = I +1 r 7: e +1 =MG V ( + 1, r +1, 0) 8: ˆx = x + I +1 e +1 9: x =SMOOTH(ˆx, A, b ) 10: end if 11: return x 5 NUMERICAL RESULTS We present numerica experiments to assess the behaviour of the p-mutigrid preconditioner for the DG and HDG discretizations. First, NS soutions of a vortex transported by uniform fow at M = 0.05 and Re = 100 are reported to show the convergence rates in space and time. The second test case deas with the soution of a two-dimensiona circuar cyinder at Re = 100 and M = 0.2, and it is used to evauate parae efficiency. 5.1 Convected vortex The test case is a modified version of the VI1 case studied in the 5 th Internationa Workshop on High Order CFD Methods [13], and consists of a two-dimensiona mesh on the domain (x, y) [0, 0.1] [0, 0.1] with periodic boundary conditions on each side. See the onine web page for information about the fow initiaization. Here the set of NS equations are soved instead of the Euer equations. Numerica experiments have been performed to assess the output error, both in space and time. The meshes were obtained in a structured-ike manner using reguar quadriateras, starting from 2 2 up to The poynomia spaces empoyed were P 4, P 5 and P 6. The L 2 state error was computed reative to the soution on a , P 6 space discretization, after one convective period T. The contour pot of the soutions at the initia and fina states are shown in Figure 1. Tabe 1 reports space discretization errors. The tests were performed using a very sma time step size, T/ t = 4000, and the ESDIRK3 scheme to ensure a negigibe time discretization error. An absoute toerance on the non-inear system of 10 10, and a reative toerance of 10 5 on the GMRES, were empoyed to ensure a ow time discretization error, Even though DG suffers ess than HDG of pre-asympthotic behaviour on such a smooth soution, both the impementations show comparabe error eves and converge with the theoretica convergence rates for every poynomia approximation shown. As a consequence of such anaysis, and considering that both the DG and HDG impementations share the same code base, we wi consider ony the CPU time as a measure of the time-to-soution efficiency. 7

8 Figure 1: Convected vortex at Re = 100, M = Mach number contours. Soution at t = 0 (eft) and t = T (right). DG HDG order grid err L2 k err L2 k E E E E P E E E E E E E E E E P E E E E E E E E E E P E E E E E E Tabe 1: L 2 soution error. Laminar vortex test case at Re = 100, M = Convergence rates for the DG and HDG discretizations. The convergence rates of the ESDIRK3 time integration scheme are aso reported in Tabe 2, and have been obtained using the 16 16, P 6 space discretization. A arge non-dimensiona time step of t = 0.1, was seected to assess the efficiency of the preconditioning strategies (BJ, LJ, ILU(0) and p-mg). A three-eve Fu mutigrid V-Cyce was empoyed, with parameters determined empiricay for best convergence. The coarser space discretizations were buit using P 1 and P 3, whie GMRES smoothers were used in each eve. 10 iterations of BJ-preconditioned GMRES smoothers have been empoyed for 8

9 DG HDG T/ t err L2 k err L2 k E E E E E E E E E E E E Tabe 2: Laminar vortex test case at Re = 100, M = Time convergence rates for the DG and HDG discretizations and ESDIRK3 scheme. DG HDG Prec. CPU time IT a ρ a CPU time IT a ρ a ILU 6.67E E LJ 4.12E E BJ 3.63E E p-mg 2.66E E Tabe 3: Performance of the preconditioners using DG and HDG discretizations. Laminar vortex test case at Re = 100, M = 0.05, discretized on a mesh with P 6 poynomias, T/ t = 10, ESDIRK3 scheme. IT a stands for the average number of GMRES iterations whie ρ a the average convergece rate. a the eves except from the coarsest, where 30 ILU 0 -preconditioned GMRES iterations were performed. Such a setting was found adequate for a of the tests presented in this work, and for both space discretizations. The numerica experiments are reported in Tabe 3, which gives the CPU time, the average and maximum number of GMRES iterations though the time integration, as we as the average convergence rate (CR). The CR is defined as ρ = (r IT /r 0 ) 1/IT with IT the number of iteratons, r 0 and r IT the residuas at the first and IT th iteration respectivey. We see that the use of a p-mutigrid preconditioner reduces consideraby the number of outer GMRES iterations both for the DG and the HDG space discretizations, but ony in the DG case is this gain refected on the CPU time. In fact, the costs of condensing-out the eement-interior variabes before the soution of the system, as we as the back-sove for their evauation using the face unknowns after the inear sove, increase the amount of oca operations and hide the benefits of having a faster goba sover in HDG. 5.2 Circuar cyinder The aminar fow around a circuar cyinder at Mach number M = 0.2 and Reynods number Re = 100 has been soved on a grid made by N e = 960 mesh eements using a P 6 space discretization. The time scheme empoyed is ESDIRK3. Figure 2 shows a snapshot of the Mach number contours. The soution accuracy was assessed in comparison with iterature data [14]. To this end, Tabe 4 shows the averaged drag, ift coefficients and Strouha number (C d, C, St) for severa tempora refinements on the same grid. The 9

10 Figure 2: Laminar fow around a circuar cyinder at Re = 100, M = 0.2. Mach number contours. (U/L) t C d C St e e e e e Tabe 4: Laminar vortex test case at Re = 100, M = Time convergence rates for the DG and HDG discretizations and ESDIRK3 scheme. coefficients were obtained by averaging a fuy deveoped soution over ten shedding periods. An overa good agreement has been found, and tempora convergence can be seen by using a non-dimensiona time step of t The numerica experiments are thus performed using t = 0.25 to maximize the advantage of using an impicit scheme for such type of probem. The parae performance of the p-mutigrid preconditioner strategy introduced in the previous section is assessed by considering the effects of domain decomposition. To evauate the CPU time, a fuy-deveoped soution is advanced in time for 10 time steps to compute the average number of GMRES iterations and the convergence rates during the non-inear soution. The computations are performed using 1 to 48 cores (NP) on a patform based on Inte Xeon X5650 processors aranged in a two-processor (12 cores) per-node fashion. A fixed reative toerance of 10 6 to stop the GMRES sover is used, as we as an absoute toerance of 10 5 for the Newton-Raphson method. Tabes 5 and 6 report the resuts of the computations. For the ILU 0 preconditioner, an increase in the number of GMRES iterations is observed for both the DG and HDG space discretizations with an increasing number of cores. This behavior is attributed to the incompete ower-upper factorization, which is performed on the squared, partition-wise bock of the iteration matrix. In this case, the preconditioner effectiveness naturay decreases as NP grows, as the amount of off-diagona bocks negected increases with NP. On the other hand, by tuning the parameters of the mutigrid preconditioner it is possibe to achieve an idea agorithmic scaabiity, i.e. the number of iterations required to sove 10

11 DG HDG NP CPU time IT a ρ a CPU time IT a ρ a E E E E E E E E E E Tabe 5: Performance of the ILU 0 preconditioner using DG and HDG discretizations. DG HDG NP CPU time IT a ρ a CPU time IT a ρ a E E E E E E E E E E Tabe 6: Performance of the p-mg preconditioner using DG and HDG discretizations. the system doesn t grow by partitioning the domain. This consideration hods true for both the DG and HDG discretizations. In terms of CPU time, in a the runs the p-mg preconditioned DG sover outperforms both the ILU 0 -preconditioned DG sover as we as HDG. In particuar, the computationa time of HDG is sti higher despite fewer inear iterations required on average to reach the same eves of accuracy. 6 CONCLUSIONS An inherited p-mutigrid strategy has been proposed and assessed in the context of DG and HDG discretizations. The agorithm has been empoyed as a preconditioner for an FGMRES sover. Compared to standard singe-grid preconditioners, the approach is abe to reduce significanty the number of iterations required for the inear system soution. This reduction is refected in the overa CPU time ony in the case of DG. For HDG, the static condensation and back-sove operations are expensive and hide the benefits of using such a strategy. Future works wi be devoted to assess the infuence of different mutigrid settings, as we as to extend the current work by considering stiff three-dimensiona cases. REFERENCES [1] M. Francioini, A. Criveini, and A. Nigro. On the efficiency of a matrix-free ineary impicit time integration strategy for high-order discontinuous Gaerkin soutions of incompressibe turbuent fows. Computers & Fuids, 159: , [2] J. S. Hesthaven and T. Warburton. Noda discontinuous Gaerkin methods: agorithms, anaysis, and appications. Springer Science & Business Media,

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