Vector and scalar penalty-projection methods

Numerical Flow Models for Controlled Fusion - April 2007 Vector and scalar penalty-projection methods for incompressible and variable density flows Philippe Angot Université de Provence, LATP - Marseille with M. Jobelin [PhD, 2006], J.-C. Latché J.-P. Caltagirone and P. Fabrie Porquerolles, April 19-2007

Motivations and objectives Work focusing on the constraint of free divergence How to deal efficiently with the free-divergence constraint with fractional-step methods (prediction-correction steps)? How to circumvent the major drawbacks of the usual projection methods including a scalar correction step for the Lagrange multiplier with solution of a Poisson-type equation? Example of fluid-type models with pressure as Lagrange multiplier solution of unsteady incompressible Navier-Stokes equations in the primitive variables (velocity and pressure) Introduction 2

Outlines 1 Projection methods for incompressible flows 2 Scalar penalty-projection methods 3 Vector penalty-projection methods 4 Conclusion and perspectives Introduction 3

Outlines 1 Projection methods for incompressible flows Non-homogeneous incompressible flows Semi-implicit method (linearly implicit) Fractional-step and projection methods 2 Scalar penalty-projection methods 3 Vector penalty-projection methods 4 Conclusion and perspectives Projection methods for incompressible flows 4

Navier-Stokes problem for incompressible flows Incompressible and variable density flows of Newtonian fluids Navier-Stokes equations with mixed boundary conditions : Dirichlet on Ω D and open (outflow) B.C. on Ω N ϱ + u ϱ = 0 in Ω ]0, T [ t [ ] u ϱ + (u )u τ (u) + p = f in Ω ]0, T [ t u = 0 in Ω ]0, T [ u = u D on Ω D ]0, T [ pn + τ (u) n = f N on Ω N ]0, T [ ϱ = ϱ 0, and u = u 0 in Ω {0} τ (u) = [µ( u + u T )], or µ u (for a constant viscosity) For an homogeneous fluid with constant density, we set ϱ = 1. Projection methods for incompressible flows Non-homogeneous incompressible flows 5

Semi-implicit method (linearly implicit) Semi-discretization in time with low or high-order BDF schemes for all n IN such that (n + 1)δt T : D ϱ n+1 + (ū,n+1 ϱ n+1) = 0 in Ω δt ( ) Dū ϱ n+1 n+1 + (ū,n+1 )ū n+1 δt τ (ū n+1 ) + p n+1 = f n+1 in Ω ū n+1 = 0 ū n+1 = u n+1 D p n+1 n + τ (ū n+1 ) n = f n+1 N in Ω on Ω D on Ω N Resolution of an elliptic problem in space (at each time step) by methods of finite elements, finite volumes, DGM... Projection methods for incompressible flows Semi-implicit method (linearly implicit) 6

Semi-implicit method : coupled solver Algebraic formulation of the Navier-Stokes system inf-sup stable discretization in space or with stabilization... β q δt M ϱu n+1 + A(U n, U n 1 )U n+1 + B T P n+1 = F(U n, U n 1, f n+1, f n+1 N, un+1 D ) BU n+1 = G(u n+1 D ) Projection methods for incompressible flows Semi-implicit method (linearly implicit) 7

Semi-implicit method : coupled solver Algebraic formulation of the Navier-Stokes system inf-sup stable discretization in space or with stabilization... β q δt M ϱu n+1 + A(U n, U n 1 )U n+1 + B T P n+1 = F(U n, U n 1, f n+1, f n+1 N, un+1 D ) BU n+1 = G(u n+1 D ) Saddle-point problem at each time step : efficient solver? Fully-coupled solver with efficient multigrid preconditioner : Kortas, PhD 1997 - Cihlář and Angot, 1999 Augmented Lagrangian method with iterative Uzawa algorithm : Fortin and Glowinski, 1983 - Khadra et al., IJNMF 2000 Projection methods using the Helmoltz-Hodge-Leray decomposition of L 2 (Ω) d Projection methods for incompressible flows Semi-implicit method (linearly implicit) 7

Fractional-step and projection-type methods Projection scheme with scalar pressure-correction [Chorin, 1968 - Temam, 1969 - Goda, 1979 - Van Kan, 1986] See recent review : [Guermond, Minev, Shen, CMAME 2006] Prediction ( step ) Dũ ϱ n+1 n+1 + (u,n+1 )ũ n+1 τ (ũ n+1 ) + p n = f n+1 δt ũ n+1 = u n+1 D on Ω D p n n + τ (ũ n+1 ) n = f n+1 N on Ω N Projection step β q ϱ n+1 un+1 ũ n+1 + φ = 0 δt φ n = 0 on Ω D (necessary since u n = ũ n) φ = 0 on Ω N (sufficient by ( orthogonal ) projection onto H) δt u n+1 = 0 φ = β ϱn+1 q ũ n+1 Pressure-correction step : φ pressure increment p n+1 = p n + φ Projection methods for incompressible flows Fractional-step and projection methods 8

Fractional-step and projection-type methods Algebraic formulation for the Navier-Stokes system β q δt MϱŨn+1 + AŨn+1 + B T P n = F L ϱ Φ = β ( ) q δt BŨn+1 G P n+1 = P n + Φ M ϱ U n+1 = M ϱ Ũ n+1 + δt β q B T Φ Projection methods for incompressible flows Fractional-step and projection methods 8

Fractional-step and projection-type methods Major drawbacks of the incremental projection methods Time order of the splitting error? i.e. error between the numerical solutions of the implicit (or semi-implicit) method and the fractional-step method φ n = 0 on Ω D existence of an artificial pressure boundary layer in space φ = 0 on Ω N convergence in time and space spoiled for outflow boundary conditions : splitting error varying like O(δt 1/2 ) (pressure) and no more negligible (for both velocity and pressure) with respect to the time and space discretization error Pressure-correction step strongly dependent on density and viscosity for non-homogeneous flows convergence very poor for large ratios of ϱ 1000. Projection methods for incompressible flows Fractional-step and projection methods 8

Numerical tests Green-Taylor vortices : Navier-Stokes with Dirichlet B.C. Velocity error (discrete L 2 (0, T ; L 2 (Ω) d ) norm) versus time step δt 1e-2 norme L2 erreur vitesse 1e-3 1e-4 1e-5 1e-6 semi implicite 1e-7 1e-4 1e-3 1e-2 1e-1 Pas de temps Time convergence in O(δt 2 ) Stagnation threshold = space discretization error in O(h 2 ) Projection methods for incompressible flows Fractional-step and projection methods 9

Numerical tests Green-Taylor vortices : Navier-Stokes with Dirichlet B.C. Velocity error (discrete L 2 (0, T ; L 2 (Ω) d ) norm) versus time step δt 1e-2 norme L2 erreur vitesse 1e-3 1e-4 1e-5 1e-6 semi implicite incrementale 1e-7 1e-4 1e-3 1e-2 1e-1 Pas de temps Time convergence in O(δt 2 ) Stagnation threshold = space discretization error in O(h 2 ) Projection methods for incompressible flows Fractional-step and projection methods 9

Outlines 1 Projection methods for incompressible flows 2 Scalar penalty-projection methods Penalty-projection methods Numerical experiments Analysis of the scalar penalty-projection method for the Stokes problem 3 Vector penalty-projection methods 4 Conclusion and perspectives Scalar penalty-projection methods 10

Penalty-projection methods [Jobelin et al., JCP 2006] : prediction step with augmented Lagrangian for r > 0 and consistent projection step by scalar pressure-correction [Shen, 1992] : r = 1/δt 2 with a different correction of pressure [Caltagirone and Breil, 1999] : r > 0 with a singular projection operator... Scalar penalty-projection methods Penalty-projection methods 11

Penalty-projection methods Penalty-prediction ( step : augmentation) parameter r 0 Dũ ϱ n+1 n+1 + (u,n+1 )ũ n+1 τ (ũ n+1 ) δt r ( ũ n+1) + p n = f n+1 ũ n+1 = u n+1 D on Ω D p n n + τ (ũ n+1 ) n = f n+1 N on Ω N Projection step β q ϱ n+1 un+1 ũ n+1 + φ = 0 δt φ n = 0 on Ω D (necessary since u n = ũ n) φ = 0 on Ω N (sufficient by( orthogonal ) projection onto H) δt u n+1 = 0 φ = β ϱn+1 q ũ n+1 Pressure-correction step : φ consistent pressure increment p n+1 = p n r ũ n+1 + φ = p n+1 + φ Scalar penalty-projection methods Penalty-projection methods 11

Penalty-projection methods Algebraic formulation for the Navier-Stokes system β ( ) q δt MϱŨn+1 + rb T M 1 pl BŨ n+1 G + B T P n +AŨ n+1 = F L ϱ Φ = β ( q δt ) BŨ n+1 G P n+1 = P n + rm 1 pl ( BŨ n+1 G M ϱ U n+1 = M ϱ Ũ n+1 + δt β q B T Φ ) + Φ Preconditioning the prediction step by one iteration of augmented Lagrangian and consistent scalar projection r = 0 : incremental projection method Scalar penalty-projection methods Penalty-projection methods 12

Penalty-projection methods Algebraic formulation for the Navier-Stokes system β ( ) q δt MϱŨn+1 + B T (rm 1 pl BŨ n+1 G + P n ) }{{} P n+1 +AŨ n+1 = F L ϱ Φ = β ( ) q BŨ n+1 G δt ( ) P n+1 = P n + rm 1 pl BŨ n+1 G +Φ }{{} P n+1 M ϱ U n+1 = M ϱ Ũ n+1 + δt B T Φ β q Preconditioning the prediction step by one iteration of augmented Lagrangian and consistent scalar projection r = 0 : incremental projection method Scalar penalty-projection methods Penalty-projection methods 12

Penalty-projection methods Direct implementation with the algebraic formulation or not... Why? Continuous term corresponding to the penalization T p = r Ω u v But T p does not generally vanish for a velocity field with zero discrete divergence! It depends on the spatial discretization... Hence Introduction of an additional error due to the space discretization Error which increases with the augmentation parameter r > 0 Scalar penalty-projection methods Penalty-projection methods 13

Numerical experiments Green-Taylor vortices : Navier-Stokes with Dirichlet B.C. Velocity error (discrete L 2 (0, T ; L 2 (Ω) d ) norm) versus time step δt 1e-2 norme L2 erreur vitesse 1e-3 1e-4 1e-5 1e-6 semi implicite incrementale 1e-7 1e-4 1e-3 1e-2 1e-1 Pas de temps Time convergence in O(δt 2 ) Stagnation threshold = space discretization error in O(h 2 ) Scalar penalty-projection methods Numerical experiments 14

Numerical experiments Green-Taylor vortices : Navier-Stokes with Dirichlet B.C. Velocity error (discrete L 2 (0, T ; L 2 (Ω) d ) norm) versus time step δt 1e-2 norme L2 erreur vitesse 1e-3 1e-4 1e-5 1e-6 semi implicite incrementale r=1 1e-7 1e-4 1e-3 1e-2 1e-1 Pas de temps Time convergence in O(δt 2 ) Stagnation threshold = space discretization error in O(h 2 ) Scalar penalty-projection methods Numerical experiments 14

Numerical experiments Green-Taylor vortices : Navier-Stokes with Dirichlet B.C. Velocity error (discrete L 2 (0, T ; L 2 (Ω) d ) norm) versus time step δt 1e-2 norme L2 erreur vitesse 1e-3 1e-4 1e-5 1e-6 semi implicite incrementale r=1 r=10 1e-7 1e-4 1e-3 1e-2 1e-1 Pas de temps Time convergence in O(δt 2 ) Stagnation threshold = space discretization error in O(h 2 ) Scalar penalty-projection methods Numerical experiments 14

Numerical experiments Green-Taylor vortices : Navier-Stokes with Dirichlet B.C. Velocity error (discrete L 2 (0, T ; L 2 (Ω) d ) norm) versus time step δt 1e-2 norme L2 erreur vitesse 1e-3 1e-4 1e-5 1e-6 semi implicite incrementale r=1 r=10 r=100 1e-7 1e-4 1e-3 1e-2 1e-1 Pas de temps Time convergence in O(δt 2 ) Stagnation threshold = space discretization error in O(h 2 ) Scalar penalty-projection methods Numerical experiments 14

Numerical experiments Green-Taylor vortices : Navier-Stokes with Dirichlet B.C. Pressure error (discrete L 2 (0, T ; L 2 (Ω)) norm) versus time step δt norme L2 erreur pression 1e-1 1e-2 1e-3 1e-4 1e-5 semi implicite incrementale r=1 r=10 r=100 1e-6 1e-4 1e-3 1e-2 1e-1 Pas de temps Stagnation threshold = space discretization error in O(h 2 ) Scalar penalty-projection methods Numerical experiments 15

Numerical experiments Artificial pressure boundary layer : Stokes with Dirichlet B.C. on a disk incremental projection penalty-projection r=1 p h p L (Ω) = 1.5 10 2 p h p L (Ω) = 2.8 10 3 Scalar penalty-projection methods Numerical experiments 16

Numerical experiments Artificial pressure boundary layer : Stokes with Dirichlet B.C. on a disk penalty-projection r=100 p h p L (Ω) = 2.8 10 4 implicit scheme p h p L (Ω) = 1.8 10 4 Scalar penalty-projection methods Numerical experiments 16

Numerical experiments Stokes with open boundary condition at a channel outflow Velocity error (discrete L 2 (0, T ; L 2 (Ω) d ) norm) versus time step δt Norme L2 erreur vitesse 1e-1 1e-2 1e-3 1e-4 1e-5 1e-6 1e-7 1e-8 semi implicite incrementale r=1 r=10 r=100 r=1000 r=10000 1e-9 1e-4 1e-3 1e-2 1e-1 Pas de temps Time convergence in O(δt 2 ) Stagnation threshold = space discretization error in O(h 2 ) Scalar penalty-projection methods Numerical experiments 17

Numerical experiments Stokes with open boundary condition at a channel outflow Pressure error (discrete L 2 (0, T ; L 2 (Ω)) norm) versus time step δt 1e+0 norme L2 erreur pression 1e-1 1e-2 1e-3 1e-4 1e-5 semi implicite incrementale r=1 r=10 r=100 r=1000 r=10000 1e-6 1e-4 1e-3 1e-2 1e-1 Pas de temps Stagnation threshold = space discretization error in O(h 2 ) Scalar penalty-projection methods Numerical experiments 18

Numerical experiments Navier-Stokes open flow around a circular cylinder Taylor-Hood P2-P1 finite elements Scheme 2nd order in time Reynolds number : Re = 100 (µ = 0.01) Unstructured mesh with 4200 elements Complete study of this configuration : see Khadra et al., IJNMF 2000 with FVM (MAC mesh) and iterative augmented Lagrangian algorithm. Splitting error varying as O( 1 r ) Scalar penalty-projection methods Numerical experiments 19

Numerical experiments Navier-Stokes open flow around a circular cylinder Norme L2 erreur de fractionnement 1e-1 1e-2 1e-3 1e-4 Pas de temps = 0.05 s Pas de temps = 0.02 s Pas de temps =0.01 s Pas de temps = 0.005 s 1e-5 1e+0 1e+1 1e+2 1e+3 paramètre de pénalisation Splitting error varying as O( 1 r ) Scalar penalty-projection methods Numerical experiments 19

Theoretical analysis for Stokes-Dirichlet problem Unsteady homogeneous model problem with ϱ = 1 Ω connected and bounded open domain of IR d, Lipschitz boundary Ω, T > 0 u u + p = f dans Ω ]0, T [ t u = 0 dans Ω ]0, T [ u = 0 sur Ω ]0, T [ u(x, 0) = u 0 (x) dans Ω u denotes the velocity vector, p the pressure field. For f L 2 (0, T ; L 2 (Ω) d ) and u 0 V given, there exists a unique solution u L (0, T ; H) L 2 (0, T ; V ) and p L 2 (0, T ; L 2 0 (Ω)) See [Temam, 1979 - Girault and Raviart, 1986] Scalar penalty-projection methods Analysis of the scalar penalty-projection method 20

Theoretical analysis for Stokes-Dirichlet problem [Shen, 1992-95 - Guermond, 1996] : standard projection (pressure-correction form) [Guermond and Shen, 2003] : standard projection (velocity-correction) [Guermond and Shen, 2004] : rotational variant of [Timmermans et al., 1996] [Angot, Jobelin, Latché, Preprint 2006] : penalty-projection Analysis for small values of the penalty parameter r Theorem (Splitting error - fully discrete case in time and space) Energy estimates of splitting errors compared to Euler implicit scheme there exists c = c(ω, T, f, u 0, h) > 0 such that : for 1 n N, [ n δt e k 2 0 k=0 [ n δt ẽ k 2 0 k=0 ] 1 [ 2 n + k=0 ] 1 [ 2 n + k=0 δt ẽ k 2 0 ] 1 2 δt ɛ k 2 0 ] 1 2 c min(δt 2, δt3/2 r 1/2 ) c max(1, 1 r 1/2 )δt3/2. Scalar penalty-projection methods Analysis of the scalar penalty-projection method 21

Theoretical analysis for Stokes-Dirichlet problem Analysis for large values of the penalty parameter r Theorem (Splitting error - fully discrete case in time and space) Energy estimates of splitting errors compared to Euler implicit scheme there exists c = c(ω, T, f, u 0, h) > 0 such that : for 1 n N, [ n e n 0 + ẽ n 0 + δt ẽ k 2 0 [ n δt e k 2 0 k=0 ɛ n h [ n δt ɛ k 2 0 k=0 k=0 ] 1 [ 2 n + ] 1 2 k=0 δt ẽ k 2 0 ] 1 2 ] 1 2 c δt 1 2 r c δt r c 1 r 1/2 c 1 r. Scalar penalty-projection methods Analysis of the scalar penalty-projection method 22

Theoretical analysis for Stokes-Dirichlet problem [ n δt ẽ k 2 0 k=0 ] 1 2 : velocity splitting error ẽ n = ū n ũ n discrete L 2 (0, T ; L 2 (Ω) d ) norm versus time step δt 1e-1 1e-2 1e-3 1e-4 r = 0 r = 1 r = 5 r = 10 r = 50 r = 100 r = 500 r = 1000 1e-5 1e-6 1e-7 1e-8 1e-4 1e-3 1e-2 1e-1 Pas de temps Scalar penalty-projection methods Analysis of the scalar penalty-projection method 23

Theoretical analysis for Stokes-Dirichlet problem [ n δt ẽ k 2 0 k=0 ] 1 2 : velocity splitting error ẽ n = ū n ũ n discrete L 2 (0, T ; L 2 (Ω) d ) norm versus penalty parameter r > 0 1e-2 1e-3 1e-4 Pas de temps: 0.1 Pas de temps: 0.05 Pas de temps: 0.025 Pas de temps: 0.0125 Pas de temsp: 0.00625 Pas de temps: 3.125e-3 Pas de temps: 1.563e-3 Pas de temps: 7.813e-4 Pas de temps: 3.906e-4 1e-5 1e-6 1e-7 1e-8 1e+0 1e+1 1e+2 1e+3 Paramètre de pénalisation Scalar penalty-projection methods Analysis of the scalar penalty-projection method 23

Theoretical analysis for Stokes-Dirichlet problem [ n δt ɛ k 2 0 k=0 ] 1 2 : pressure splitting error ɛ n = p n p n discrete L 2 (0, T ; L 2 (Ω)) norm versus penalty parameter r > 0 1e-1 1e-2 1e-3 Pas de temps: 0.1 Pas de temps: 0.05 Pas de temps: 0.025 Pas de temps: 0.0125 Pas de temps: 0.00625 Pas de temps: 3.125e-3 Pas de temps: 1.563e-3 Pas de temps: 7.813e-4 Pas de temps: 3.906e-4 1e-4 1e-5 1e-6 1e-7 1e+0 1e+1 1e+2 1e+3 Paramètre de pénalisation Scalar penalty-projection methods Analysis of the scalar penalty-projection method 23

Outlines 1 Projection methods for incompressible flows 2 Scalar penalty-projection methods 3 Vector penalty-projection methods New class of vector penalty-projection methods Analysis of the vector penalty-projection method for the Stokes problem 4 Conclusion and perspectives Vector penalty-projection methods 24

New class of vector penalty-projection methods Work in progress with Caltagirone and Fabrie A two-parameter family with vector projection step Penalty-prediction ( step : augmentation) parameter r 0 Dũ ϱ n+1 n+1 + (u,n+1 )ũ n+1 τ (ũ n+1 ) δt r ( ũ n+1) + p n = f n+1 ũ n+1 = u n+1 D on Ω D, and p n n + τ (ũ n+1 ) n = f n+1 N on Ω N p n+1 = p n r ũ n+1 Vector [ projection ( step : penalty parameter ) 0 < η δt ] η ϱ n+1 βq δt ûn+1 + (u,n+1 )û n+1 τ (û n+1 ) ( û n+1) = ( ũ n+1) û n+1 = 0 on Ω D, and ( p n+1 p n )n + τ (û n+1 ) n = 0 on Ω N Correction step : u n+1 = ũ n+1 + û n+1, and p n+1 = p n+1 1 η un+1 Vector penalty-projection methods New class of vector penalty-projection methods 25

New class of vector penalty-projection methods Why does it work well? Very nice conditioning properties of the vector penalty-correction step as η 0 since the right-hand side becomes more and more adapted to the left-hand side operator : only a few iterations (2 or 3) of MILU0-BiCGStab solver whatever the mesh step h! Original boundary conditions not spoiled through a scalar projection step Penalty-correction step quasi-independent on the density or viscosity when η 0 for non-homogeneous incompressible flows It is only an approximate projection scheme since u n+1 0 But both the velocity divergence and the splitting error can be made negligible with respect to the time and space discretization errors when η 0 cheap method for small values of r < 1 and η 10 15 (zero machine) Vector penalty-projection methods New class of vector penalty-projection methods 26

Analysis for Stokes-Dirichlet problem with r = 0 Error estimates of the one-parameter family : r = 0 and 0 < η δt Theorem (Splitting error in the semi-discrete case) Energy estimates of splitting errors w.r. to Euler implicit scheme there exists C = C(Ω, T, f, u 0 ) > 0 such that : for 1 n N, [ n e n 0 + δt e k 2 0 [ n δt e k 2 0 k=0 k=0 ] 1 2 ] 1 2 C η δt C ηδt [ n η ɛ n 0 + δt ɛ k 2 0 k=0 ] 1 2 C η u n 0 = e n 0 C max ( 1, η δt ) ηδt. Vector penalty-projection methods Analysis of the vector penalty-projection method 27

Outlines 1 Projection methods for incompressible flows 2 Scalar penalty-projection methods 3 Vector penalty-projection methods 4 Conclusion and perspectives Conclusion and perspectives 28

Conclusion Scalar penality-projection methods for incompressible flows Reduce the splitting error up to make it negligible Suppress pressure boundary layers for moderate values of r > 0 Optimal convergence for r > 0 with open boundary conditions Generalization to dilatable and low Mach number flows : see [Jobelin et al., Preprint 2006] Require efficient preconditioning for large values of r multi-level preconditioner for FVM with MAC mesh : see [Kortas, PhD 1997]. Conclusion and perspectives 29

Perspectives Vector penality-projection methods for incompressible and non-homogeneous flows Small values of r < 1 are sufficient to get a good pressure field Approximate projection with a vector correction step all the cheaper than η 0 Same convergence properties as before Vector correction step all the less dependent on density or viscosity than η 0 Other numerical experiments (in progress) Error estimates : two-parameter class, Navier-Stokes, outflow B.C., variable density flows... Conclusion and perspectives 30